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A  TREATISE 


ON 


PLANE   SURVEYING 


BY 


DANIEL  CARHART,  C.E. 

.PROFESSOR  OP  CIVIL  ENGINEERING  IN  THE  WESTERN 
UNIVERSITY  OF  PENNSYLVANIA 


48504 

GINN  &  COMPANY 

BOSTON*  •  NEW   YORK  •   CHICAGO  •  LONDON 


• 


Entered,  according  to  Act  of  Congress,  in  the  year  1887,  by 

DANIEL  CARHART 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington 

68.5 


JCfje   atfcenacutn 


PREFACE. 


rr^HIS  work,  as  its  name  indicates,  extends  over  the  field  of 
J-  plane  surveying.  It  illustrates  and  describes  the  instru- 
ments employed,  their  adjustments  and  uses ;  it  exemplifies  the 
best  methods  of  solving  the  common  problems  occurring  in 

Y  practice,  and  furnishes  solutions  for  many  special  cases  which 

?^  not  unfrequently  present  themselves.  An  experience  of  twenty 
-  years  in  the  field  and  in  technical  schools  confirms  the  opinion 
that  a  work  of  this  kind  should  be  eminently  practical ;  that 
the  student  who  desires  to  become  a  reliable  surveyor  needs 
frequently  to  manipulate  the  various  surveying  instruments  in 
the  field,  to  solve  many  examples  in  the  class-room,  and  to 
exercise  good  judgment  in  all  these  operations.  With  this  in 

^  view,  therefore,  the  different  methods  of  surveying  are  treated, 
^   directions  for  using  the  instruments  are  given,  and  these  are 

^  supplemented  by  numerous  examples  to  be  solved,  by  various 
field  exercises  to  be  performed,  and  by  many  queries  to  be 
answered. 

Chapter  I.  is  devoted  to  Chain  Surveying,  in  which  direc- 
tions  are  given  for  measuring  and  ranging  out  lines,  and 
methods  of  overcoming  obstacles,  recording  field  notes,  obtain- 
ing  areas,  and  plotting  a  chain  survey. 

Chapter  II.  treats  of  Compass  and  Transit  Surveying,  or 
when,  in  addition  to  the  chain,  an  instrument  for  measuring 
angles  is  employed.  In  this  chapter  the  compass  and  transit, 
the  solar  attachment,  the  adjustments  of  these,  and  auxiliaries 
of  the  transit,  such  as  the  stadia  wires,  gradienter,  etc.,  are 
fully  illustrated  and  described,  and  their  uses  shown.  Here 
the  various  methods  of  obtaining  the  data  requisite  to  deter- 


IV  PREFACE. 

mine  the  area,  as  well  as  the  different  methods  employed  in 
calculating  the  contents  of  laud,  are  exhibited.  Tests  of  the 
accuracy  of  a  survey  are  indicated,  numerous  methods  of  over- 
coming obstacles,  supplying  omissions,  of  ascertaining  heights 
and  distances,  of  keeping  the  field  notes,  and  of  plotting  a  sur- 
vey are  given,  while  the  uses  of  the  solar  attachment  in  deter- 
mining the  latitude  of  a  station  and  its  geographic  meridian  are 
exemplified. 

The  student  now  having  been  taught  how  to  survey  land, 
using  a  needle  instrument,  should  become  acquainted  with  the 
declination  of  the  magnetic  needle,  or  variation  of  the  compass, 
as  it  is  frequently  called.  This  subject  is  accordingly  dis- 
cussed in  Chapter  III.  Some  of  the  tables  and  much  of  the 
matter  is  taken  from  the  Reports  of  the  United  States  Coast 
and  Geodetic  Survey.  The  student  will  do  well  to  give  this 
chapter  a  careful  inspection,  examining  the  tables  and  formu- 
las and  the  directions  for  determining  the  true  meridian,  thus 
being  prepared  with  facts,  figures,  and  methods,  which  will  en- 
able him  intelligently  to  undertake  the  retracing  of  old  lines, 
as  well  as  to  establish  with  considerable  precision  his  geo- 
graphic meridian,  and  thereby  obtain  the  declination  of  the 
needle. 

Chapter  IV.  is  devoted  to  Laying  Out  and  Dividing  Up  Land. 
This  subject  is  of  more  importance  than  some  suppose,  especi- 
ally to  practitioners  in  the  older  States  of  the  Union,  and  is  here 
treated  very  fully.  The  principal  cases  are  exemplified,  and 
general  directions  and  suggestions  given,  so  that,  it  is  believed, 
with  a  thorough  knowledge  of  this  chapter,  the  student  will  be 
enabled,  without  embarrassment,  to  meet  the  requirements  of 
an  extensive  practice. 

The  description,  adjustment,  and  use  of  the  Plane  Table  form 
the  subject  of  Chapter  V.  This  instrument  is  being  employed 
more  frequently  than  formerly  in  park  surveys,  in  determining 
positions  in  harbors,  along  the  lines  of  proposed  highways,  in 
"filling  in"  large  surveys,  and 'generally  in  locating  points 
where  extreme  accuracy  is  not  required. 


PREFACE.  V 

In  Chapter  VI.  the  system  employed  by  the  government  in 
the  Survey  of  the  Public  Lands  is  set  forth.  The  description 
and  adjustment  of  the  Solar  Compass,  which  is  used  quite  ex- 
tensively in  these  surveys,  precede  au  account  of  the  origin  of 
the  system,  and  the  leading  points  in  the  "  Instructions  to  Sur- 
veyors-General "  from  the  commissioner  of  the  land  office.  A 
form  of  recording  the  notes  extracted  from  the  "  Instructions" 
is  also  given,  the  chapter  closing  with  formulas  and  a  table  for 
determining  the  inclination  of  meridians  and  deviation  of  par- 
allels. 

Chapter  VII.,  on  City  Surveying,  is  from  the  pen  of  my 
friend  and  former  colleague,  Frederic  H.  Robinson,  C.E.,  City 
Engineer  of  Wilmington,  Del.  This  subject  has  received  but 
little  notice  from  writers  on  surveying,  although  the  need  of 
some  systematic  and  practical  treatment  of  it  has  long  been 
recognized.  It  therefore  affords  me  much  pleasure  to  acknowl- 
edge my  indebtedness  to  Professor  Robinson  for  supplying  this 
want,  and  so  enhancing  the  value  of  this  publication  as  a  text- 
book. Experience  in  teaching,  and  ten  years'  practice  in  city- 
surveys  and  improvements,  eminently  qualify  him  to  speak  on 
this  important  subject  with  authority  and  in  a  manner  readily 
understood  by  students. 

The  special  instruments  needed  in  this  branch  of  surveying 
are  illustrated  and  described ;  the  adjustment  of  the  Y-level 
and  directions  how  to  level  and  to  record  the  notes  are  given ; 
more  refined  means  of  measuring  lines  are  discussed  ;  tempera- 
ture, pull,  sag,  wind,  etc.,  are  considered,  and  corrections  indi- 
cated ;  best  directions  and  width  of  streets,  together  with  the 
subject  of  grades,  sewers,  the  establishment  of  permanent 
reference  points,  and  adjusting  property  lines,  are  fully  set 
forth. 

To  my  college  classmate  and  esteemed  friend,  F.  Z.  Schel- 
lenberg,  C.E.,  Superintendent  of  Westmoreland  Coal  Co., 
Irwin,  Pennsylvania,  I  am  indebted  for  Chapter  VIII.,  on 
Mine  Surveying.  This  chapter,  though  in  general  explanatory 
of  what  is  applicable  and  peculiar  to  this  branch  of  surveying, 


vi  PREFACE. 

includes  directions  for  running  contours  and  sketching  topog- 
raphy. It  is  replete  with  suggestions  that  will  be  valued 
when,  by  the  aid  of  the  study  of  mine  workings  themselves  and 
their  ground,  illustrations  will  be  afforded  which  otherwise,  as 
drawings  alone,  cannot  readily  be  understood. 

The  Judicial  Functions  of  Surveyors,  as  given  by  Chief  Jus- 
tice Cooley,  are  set  forth  in  an  Appendix. 

Those  who  are  familiar  with  the  elegant  tables  of  logarithms 
of  numbers  and  of  trigonometrical  functions  prepared  by  Pro- 
fessor Wentworth,  will  likely  recognize  the  use  of  his  electro- 
plates, from  which  I  have  been  permitted  to  print  Tables  I., 
III.,  IV.,  and  VII.  To  him  mv  personal  acknowledgments 
are  due.  The  plates  from  which  Tables  II.,  V.,  VI.,  VIII.,  and 
IX.  are  printed  were  prepared  expressly  for  this  work.  It  is 
thought  that  the  four-place  tables  of  the  natural  trigonometri- 
cal functions  will  be  found  very  useful  in  connection  with  sur- 
veying and  engineering  operations.  They  are  believed  to  be 
correct,  having  been  very  carefully  compared  with  others  whose 
accuracy  is  unquestioned. 

In  addition  to  acknowledgments  made  elsewhere,  I  take 
pleasure  in  expressing  here  my  thanks  to  Messrs.  W.  and  L.  E. 
Gurley,  of  Troy,  New  York,  for  the  use  which  I  have  been  per- 
mitted to  make  of  their  valuable  catalogue,  in  the  description 
of  certain  instruments,  and  for  the  loan  of  several  plates  for 
the  engraving  of  instruments  ;  also  to  Messrs.  Fauth  and  Co., 
"Washington,  D.C.,  and  to  Messrs.  Heller  and  Brightly,  and 
Messrs.  Young  and  Sons,  Philadelphia,  Pa.,  for  plates  which 
they  kindly  furnished  for  the  illustration  of  the  subject. 

D.  C. 

WESTERN  UNIVERSITY  OF  PENNSYLVANIA, 
DECEMBER,  1887. 


CONTENTS. 


PAGE 

Definitions,  and  division  of  the  subject 1 


CHAPTER  I. 
CHAIN   SURVEYING. 

SECTION  I. 

INSTRUMENTS. 

Gunter's  chain „ . 8 

Two-pole  or  half  chain 8 

The  engineer's  chain 4 

The  tape  measure 4 

Marking-pins 4 

Straight  poles 4 

SECTION  II. 
CHAINING. 

How  to  chain 4 

Tallying : 5 

Error  in  chaining 6 

Chaining  on  sloping  ground 8 

Field  exercises 10 

Ranging  out  lines '. 10 

Over  a  hill ;  across  a  valley 11 

Through  a  wood 12 

Field  exercises 13 

To  set  off  a  perpendicular  from  a  line  13, 14 

To  let  drop  a  perpendicular  from  a  point  to  a  line 14, 15 

Through  a  point  to  run  a  line  parallel  to  a  given  line 16, 17 

Obstacles  to  alignment 17, 18 


Viil  CONTENTS. 

PAGE 

Obstacles  to  measurement «...  18-20 

Measurement  of  heights 20 

Examples  and  field  exercises 21 

SECTION   III. 
RECORDING  THE  FIELD  NOTES. 

By  sketch 22 

In  columns 23-25 

SECTION  IV. 
MAPPING  AND  PLOTTING. 

Instruments  useful  for  plotting  a  chain  survey 26 

Drawing-board,  T-square,  triangles,  etc 26, 27 

Scales 28 

Drawing  to  scale 29 

To  ascertain  unknown  scale 30 

SECTION  V. 
ON  AREAS,  AND  ILLUSTRATIVE  EXAMPLES. 

Areas  :  The  area  of  a  triangle 31 

"          "     rectangle 31 

"          "     parallelogram 31 

"          "     trapezoid 31 

"          "     regular  hexagon 31 

"          "          "       octagon 31 

"          "          "       polygon  32 

Table  of  areas  of  regular  polygons 33 

The  area  of  a  circle 33 

"  "    quadrant 34 

"  "    sextant 34 

"          "    circular  ring  34 

"          "    segment 34 

"    ellipse 35 

EXAMPLES  INDICATING  HOW  TO  SURVEY  LAND,  TO  PLOT  THE  SURVEY, 
AND  TO  COMPUTE  THE  AREA. 

To  survey  a  triangle 35,  36 

"          "     rectangle „ . .     37 

Examples 37 


CONTENTS.  ix 

PAGE 

To  survey  a  parallelogram 38 

"          "      trapezoid , 38 

Examples 38,  39 

To  survey,  plot,  and  compute  the  area  of  a  trapezium 39 

Examples 40 

To  survey,  plot,  and  compute  the  area  of  a  polygon,  regular  or  irreg- 
ular   40,  41 

Examples 41, 42 

To  survey,  plot,  and  compute  the  area  of  a  circle  and  a  circular  ring.     43 

"  "  "    sector  and  a  segment 43 

Examples 43 

SECTION  VI. 
OFFSETS  AND  TIE-LINES. 

Remarks  and  illustrations 44 

Rectangular  co-ordinates  and  their  application  to  the  computation  of 

areas 45 

Formula  and  rule 46 

Examples 47 

Additional  formulas  and  rule 48 

Examples 48-50 

Tie-lines  used  to  survey  a  lake  or  pond 51 

Miscellaneous  examples 51,  52 

Field  exercises 53 


CHAPTER   II. 
COMPASS  AND  TRANSIT  SURVEYING. 

SECTION  I. 
DEFINITIONS  AND  DESCRIPTION  OF  INSTRUMENTS. 

A  meridian  plane;  meridian  line;  the  magnetic  needle;  the  magnetic 

meridian 54 

The  azimuth,  bearing,  and  meridian  distance  of  a  line 56 

Horizontal  angle ;  vertical  angle ;  angle  of  elevation  ;  angle  of  depres- 
sion   55 

The  surveyor's  compass 55,  66 

To  adjust  the  compass 56 

Engraving  of  compass 57 

Caution  when  handling  the  compass 58 


X  CONTENTS. 

PAGE 

To  re-magnetize  the  needle 60 

Weight  of  compass •. 60 

The  vernier 61 

Formulas  for  determining  the  least  count 61 

How  to  space  a  vernier  for  a  given  least  count 62 

To  read  an  instrument  having  a  vernier 62 

Exercises  on  designing  verniers 62 

Engraving  of  surveyor's  transit 64 

Description  of  transit,  and  section  of  telescope  and  cross-wires 65,  66 

Sectional  view  through  the  spindle  of  transit „ .     67 

The  needle,  tangent  movements,  levels,  and  verniers 68, 69 

The  sockets,  levelling-plate,  and  tripod 70,  71 

To  adjust  the  transit ., 71-75 

To  use  the  transit - 76 

Transit  attachments  and  their  adjustments 77,  78 

The  solar  attachment 79, 80 

Engraving  of  solar  attachment  (Gurley's) 81 

The  adjustment  of  the  solar  attachment 83-85 

To  find  the  latitude  by  means  of  the  solar  attachment 85 

To  run  lines  with  the  solar  attachment 87 

Saegmuller's  solar  attachment  and  its  adjustments 88 

Engraving  of  Saegmuller's  attachment 89 

Latitude  by  means  of  Saegmuller's  attachment 91 

Latitude  by  a  circumpolar  star 91 

Table  of  refraction 92 

To  find  the  meridian  and  declination  of  the  needle,  using  the  attach- 
ment      93 

Time  and  azimuth  with  the  solar  attachment 94 

Error  in  declination  or  latitude  causing  an  error  in  azimuth 94 

Table  of  errors  in  azimuth  for  one-minute  error  in  latitude  or  declina- 
tion      95 

The  stadia  or  micrometer 95,  96 

Formula  when  stadia  is  perpendicular  to  level  line  of  sight 97 

Formula  when  line  of  sight  is  not  level 99 

Examples 100 

Gradienter 100, 101 

SECTION  II. 
BEARINGS  WITH  COMPASS,  AND  ANGLES  WITH  TRANSIT. 

To  obtain  the  bearing  of  a  line 102 

Reverse  bearing ;  local  attraction 103 


CONTENTS.  Xi 

PAGE 

Proof -bearings  and  tests  of  accuracy 104 

Suggestions  and  field  exercises 105 

To  measure  angles  with  the  transit,  by  repetition 106 

"  "  "  "       by  series ;  remark 107 

Deflection  angle ;  traversing,  or  surveying  by  the  back  angle 108-110 

To  traverse  a  road  or  a  small  stream Ill 

Problems  on  bearings  and  angles 112-115 

To  change  the  bearings  of  the  sides  of  a  survey 116 

Examples  and  field  exercises 117-119 

SECTION  III. 
PROBLEMS  ON  PERPENDICULARS  AND  PARALLELS. 

To  let  fall  a  perpendicular  from  a  given  point  to  a  line 120, 121 

To  prolong  a  line  past  an  obstacle  (various  methods) 121-124 

To  measure  a  line  past  an  obstacle  (various  methods) 124-127 

Examples  and  field  exercises 127, 128 

SECTION  IV. 
HEIGHTS  AND  DISTANCES. 

Measurement  of  accessible  heights , 129 

Examples 130 

Measurement  of  inaccessible  heights 130-132 

Examples 132, 133 

Inaccessible  distances 135 

The  three-point  problem  136-138 

Miscellaneous  problems  and  field  exercises 140,141 

SECTION  V. 
RECORDING  THE  FIELD  NOTES. 

Tabular  form 142 

By  sketch 142-144 

In  columns 147-149 

Large  plot  of  a  survey 153 

SECTION  VI. 
LATITUDES  AND  DEPARTURES. 

Definition  of  latitude  and  departure 154 

Formulas  and  examples  on  latitudes  and  departures 155 

The  traverse  table  and  explanation  of  the  same 156, 157 


Xii  CONTENTS. 

PAGB 

The  use  of  the  traverse  table  exemplified 157, 158 

Table  of  sines  and  cosines  employed  ;  examples 159 

Testing  a  survey 160 

Correcting  latitudes  and  departures 161 

"  "  "  "         sides  weighted 165 

SECTION  VII. 
SUPPLYING  OMISSIONS. 

General  observations  on  supplying  omissions  166 

Determination  of  one  side 167 

"  "  the  length  of  two  sides 169 

"  "          bearing  of  two  sides 172 

"  "         bearing  of  one  side  and  length  of  another 174 

SECTION  Vin. 
PLOTTING  A  COMPASS  OR  TRANSIT  SURVEY. 

Using  the  protractor 178, 179 

By  latitudes  and  departures 180, 181 

Using  cross-section  paper 182 

SECTION  IX. 
ON  DETERMINING  AREAS. 

Determining  the  area  of  triangles  and  parallelograms 183 

"  "  trapezoids  and  trapeziums 184 

Examples 185 

Determining  the  area  of  polygons 186 

General  method  for  determining  the  area  of  any  rectilinear  figure.  187-190 

Examples 191-193 

Method  by  total  latitudes  and  departures 193, 194 

Examples 194-196 

Determining  areas  when  offsets  are  taken  ...  196,  197 

Field  exercises 197, 198 

CHAPTER   III. 

DECLINATION   OF  THE  NEEDLE,   OR   VARIATION 
OF  THE  COMPASS. 

Irregular  changes 199 

The  diurnal  variation . .   200 


CONTENTS.  Xiii 

PAGE 

Table  for  reducing  observed  declinations 201 

The  secular  variation 201 

Line  of  no  declination,  or  agonic  line .201,  202 

Formulas  expressing  the  magnetic  declination 203 

Tables  exhibiting  annual  variation,  etc 206 

Effects  of  the  secular  change 208 

Rule  for  obtaining  bearings  when  variation  is  known 209 

Change  determined  by  old  lines 210 

Examples 211,  212 

To  obtain  the  true  bearing  of  a  line 212 

To  determine  the  declination,  by  Polaris 214 

Tables  of  elongation  and  azimuth  of  Polaris 216,  217 

To  establish  a  true  meridian  with  a  transit 218 

To  obtain  approximately  the  meridian , . . . .  219 


CHAPTER   IV. 

LAYING   OUT   AND   DIVIDING   LAND. 
SECTION   I. 

LAYING  OUT  LAND. 

Triangles  ;  examples   221-223 

Squares 223 

Rectangles  ;  examples 224,  225 

Parallelograms  ;  examples 225,  226 

Polygons  ;  examples 227,  228 

Circles  and  ellipses  ;  examples 228-231 

Additional  problems 232 

SECTION  II. 
DIVIDING  LAND. 

Triangles,  various  cases  ;  numerous  examples 232-240 

Trapezoids,  "  "  241-246 

Trapeziums,  "  "  246-252 

Polygons,  "  "  253-258 

Irregular  boundary 258 

Straightening  boundary  lines 258 

Miscellaneous  examples 259-261 


xiv  CONTENTS. 

CHAPTER  V. 
PLASE   TABLE   SURVEYING. 

PAGE 

The  plane  table  ................................................  262 

Engraving  of  plane  table  ........................................  263 

The  alidade  and  declinator  ......................................  265 

Adjustments  ...................................................  266 

Surveying  by  radiation,  progression  ...    .......................  267,  268 

"           "  intersection,  resection  ..............................  269 

The  three-point  problem  .........................................  270 

Bessel's  method  by  inscribed  quadrilateral  ----  .  ...................  270 

The  two-point  problem  ..............................  ............  271 

Practical  suggestions  ............................................  273 

Field  exercises  ....    ................................  .  .........  274 


CHAPTER   VI. 
THE   SURVEY   OF   THE   PUBLIC   LANDS. 

The  solar  compass  ..............................................  275 

The  latitude  and  declination  arcs  .................................   276 

Engraving  of  solar  compass  ......................................   277 

The  hour  arc  and  polar  axis  .......................................  279 

Principles  of  the  solar  compass  ........................  '  ...........  280 

Allowance  for  declination  ...............................    .......  281 

To  adjust  the  solar  compass  ..................................  283-285 

To  use  the  solar  compass  .......................................   285 

To  set  off  the  declination  .....................................  285,  286 

Allowance  for  refraction  ...........    ............................  287 

Table  of  refractions  in  declination  .............................  288,  289 

Explanation  of  the  table  of  refractions  ...........................   290 

Problems  in  declination  and  refraction  ........................  290,  291 

To  set  off  the  latitude  ...........................    ...............  291 

To  run  lines  with  the  solar  compass  ..............................   292 

Allowance  for  the  earth's  curvature  ....................    .........  293 

Time  of  day  by  the  sun  ...  .......................................  294 

Caution  as  to  the  false  image  ...................................  294 

Approximate  bearings,  and  time  for  using  solar  compass  ............   295 

Origin  of  the  system  for  the  survey  of  the  public  lands  .............   295 

Plan  of  a  township,  showing  sections  ...........  ,  .................  2% 


CONTENTS.  XV 

PAGE 

System  of  rectangular  surveying 297 

Conformity  to  the  meridian  required 298 

How  to  survey  section  lines  ;  plan  of  township 299 

An  instrument  operating  independently  of  needle  required 300 

The  four-pole  chain  to  be  adjusted  to  66.06  feet 301 

Process  of  chaining 302 

Marking  lines- 303 

Obstacles  in  line  ;  witness  points 304 

Establishing  corners 304 

Miscellaneous  directions 305,  306 

Quarter-section  corners  ;  witness  corner 307 

Meandering ., 307 

Base  line 308 

Principal  meridian ;  standard  parallels   809 

Auxiliary  meridians  ;  township  lines 310 

Method  of  subdividing  townships 311,  313 

Subdivision  of  sections ." 314-316 

Re-establishment  of  lost  corners 316 

Objects  required  to  be  noted 316,  317 

Specimen  field  notes 318-321 

Inclination  of  the  meridian 322,  323 

Convergency  of  meridians ;  deflection  of  range-lines 324 

Table  of  inclination  and  convergency  of  meridians 325 


CHAPTER  VII. 

CITY   SURVEYING. 

Introduction 326,  327 

SECTION  I. 
FIELD  INSTRUMENTS,  THEIR  ADJUSTMENTS  AND  GENERAL  USE. 

Transit,  engraving  of,  showing  gradienter,  etc .328,  329 

Rods,  steel  tapes,  etc. 331,  332 

Locke's  hand-level 333 

Effects  of  temperature,  sag,  and  wind  considered 334 

Corrections  for  these  ;  illustrations 335,  336 

Suggestions  on  measurement 336 

The  Y-level 337 

Engravings  of  Y-level 338,339 

Adjustments  of  the  level 340-343 


XVi  CONTENTS. 

PAGE 

To  use  the  level 844 

Engravings  of  New  York  and  Philadelphia  levelling-rods 345 

Description  of          "  "  "  "          "      347,348 

The  rod-level ;  engraving  of  same 349 

Levelling  defined 349,  350 

Curvature  and  refraction  considered 350,  351 

Exercises  ;  levelling  by  fore  and  back  sights 352,  353 

Levelling,  general  method 354,  355 

Form  of  record  ;  observations  on  bench  marks 356 

General  method  of  running  a  grade  line 357 

Office  instruments 358 

SECTION  II. 

A.  FIELD  WORK. 

Public  work ;  street  lines ;  direction  and  width  of  streets 358,  359 

The  alignment,  how  made 360 

Preservation  of  points  in  line 361 

Street  grades  ;  the  profile 362,  363 

llecord  of  levels  for  profile 365 

Marking  of  lines  and  grades 366 

Private  work 366 

Diagram  of  lot  and  block 367 

Locating  lot  corners  from  descriptions  in  deed 369 

How  to  proceed  when  obstacles  intervene 370 

B.  OFFICE  WORK. 

Public  work  ;  plans,  profiles  ;  exercise 372,  373 

Determination  of  grades , 374 

Diagram 375 

Drainage  377 

Private  work 378 

Conclusion  ;  books 378,  379 


CHAPTER  VIII. 

MINE    SURVEYING. 

Its  purpose „ 380 

Instruments  employed  in  making  the  alignment 381,  382 

Stations,  where  made,  and  how  designated 382 

Angles  ;  initial  course ;  reduced  courses , .  . .  382 


CONTENTS.  XVli 

PAGE 

Curves,  how  laid  out 383, 384 

The  record 385 

Mapping  and  plotting ;  inside  and  outside  work 385, 386 

The  use  of  the  level  and  rod 387, 388 

The  use  of  the  vertical  circle 388,  389 

Transference  of  points  from  surface  to  bottom  of  shaft 389 

Special  appliances  for  taking  courses  on  pitches  at  high  angles 390 

The  hanging  compass  and  hanging  clinometer 390,  391 

Topography,  its  use,  how  taken 391,  393 

Employment  of  the  plane  table 393 

Contour  lines 393 

Angular  cross-sectioning 394 

Transit  as  first  made' in  1831  ..  , 396 


APPENDIX. 
The  judicial  functions  of  surveyors 897-411 

TABLES. 

Logarithms  of  numbers 1-19 

Approximate  equation  of  time 20 

Logarithms  of  trigonometric  functions 21-49 

For  determining  with  greater  accuracy  than  by  the  preceding 50,  51 

Lengths  of  degrees  of  latitude  and  longitude 52 

Miscellaneous  formulas,  and  equivalents  of  metres,  chains,  and  feet,  53 

Traverse 54-61 

Natural  sines  and  cosines 62-70 

Natural  tangents  and  cotangents 71-79 

For  tables  in  the  body  of  the  book,  see  "  Tables  "  in  Index. 


SURVEYING. 


DEFINITIONS,  AND  DIVISION  OP  THE  SUBJECT. 

1.  Surveying  is  the  art  of  determining  and  delineating  the 
relative  position  of  points  upon  the  surface  of  the  earth.     It 
consists  principally  in  measuring,  laying  out,  and  dividing  land  ; 
in  establishing  lost  positions  ;   in  the  measurement  of  heights 
and  distances ;  and  in  the  graphical  representation  of  the  pecu- 
liarities of  any  part  of  the  earth's  surface. 

2.  It  mav  be  divided  into  two  parts :   PLANE  SURVEYING  and 
GEODETIC  SURVEYING. 

In  Plane  Surveying  the  spherical  form  of  the  earth  is  neg- 
lected ;  in  other  words,  the  portion  of  the  earth  included  in  the 
survey  is  regarded  as  a  horizontal  plane.  This  may  be  done 
without  sensible  error  where,  as  in  ordinary  land  surveying,  the 
operations  are  limited  to  surfaces  of  small  extent. 

In  Geodetic  Surveying  the  shape  of  the  earth  is  regarded, 
since  the  surfaces  under  consideration  are  so  extensive,  as  in 
the  United  States  Coast  and  Geodetic  Surveys,  sensible  errors 
would  otherwise  arise. 

RKMARK.  The  spherical  excess  of  a  spherical  triangle,  each 
of  whose  sides  is  one  mile,  is  less  than  six-thousandths  of  a 
second.  The  excess  amounts  to  only  one  second  for  an  area  of 
75.5  square  miles,  each  side  of  the  equilateral  triangle  being 
then  about  13  miles. 


2  SURVEYING. 

3.    In  the  following  pages  Plane  Surveying  only  will  be  con- 
sidered, and  the  subject  treated  under  the  following  heads : 

CHAIN  SURVEYING. 

COMPASS  AND  TRANSIT  SURVEYING. 

PLANE  TABLE  SURVEYING. 

GOVERNMENT  SURVEYING 

CITY  SURVEYING. 

MINE  SURVEYING. 

In  Plane  Surveying  there  are  usually  three  operations : 

1.  The  Field  Work. 

2.  The  Graphical  Representation,  or  Plot 

3.  The  Computation. 


CHAPTER  I. 


CHAIN  SUKVEYING, 


SECTION  I. 

INSTRUMENTS. 

4.  Chain  Surveying  has  chiefly  for  its  object  the  determina- 
tion of  areas  from  data  obtained  by  direct  measurement   of 
distances  between  points.     The  instruments  needed  are  there- 
fore simply  those  for  measuring  lines. 

5.  Gunter's  Chain,  so  called  from  its  inventor,  is  generally 
used  for  this  purpose.     It  is  made  of  iron  or  steel  wire,  is  66 
feet  in  length,  and  divided  into  100  links,  so  that  each  link,  with 
half  the  rings  connecting  it  with  the  adjoining  links,  is  seven 
and  ninety-two  hundredths  inches  (7.92),  or  one-hundredth  of  a 
chain.    Swivels  are  inserted  to  keep  it  from  twisting,  and  every 
tenth  link  has  a  metallic  mark  attached,  so  that  the  number  of 
tens  from  either  end  is  readily  ascertained.     Its  advantages  in 
surveying  farms  or  fields  are  apparent :  there  being  4840  square 
yards  in  an  acre,  and  the  chain  22  yards  long,  a  square  chain 
will  contain  one-tenth  of  an  acre  ;  or,  there  being  10,000  square 
links  in  a  square  chain,  which  is  one-tenth  of  an  acre,  100,000 
square  links  are  equivalent  to  an  acre.     Hence,  if  the  area  of  a 
field  is  calculated  in  links,  the  area  is  at  once  shown  in  acres, 
by  cutting  off  the  last  five  figures.     If  the  area  is  found  in 
chains,  then  since  there  are  ten  square  chains  in  an  acre,  the 
area  is  given  in  acres  by  cutting  off -the  last  figure. 

6.  A  Two-Pole,  or  Half-Chain  is  sometimes  used  instead  of 
Gunter's  Chain.      It  is  quite  convenient  for  measuring  lines 
where  the  ground  is  rough  and  hilly. 


4  PLANE   SURVEYING. 

7.  The  Engineer's  Chain  is  used  in  surveying  railroads  and 
canals,  and  generally  where  extensive  line  surveys  are  being 
conducted  ;  hence  not  unfrequently  it  is  employed  in  connection 
with  these  surveys,  as  well  as  otherwise,  in  determining  areas. 
It  is  100  feet  in  length,  and  is  divided  into  100  links,  every 
tenth  link  being  marked  by  a  piece  of  brass,  as  in  the  four-pole 
chain. 

8.  The  Tape  Measure  is  very  convenient  for  taking  offsets 
in  a  survey,  for  measuring  the  boundaries  of  city  lots,  cross- 
sectioning  in  railroad  work,   etc.      Tapes  are  "metallic,"  or 
steel,  and  made  of  various  lengths,*  —  50  feet  or  100  feet  are 
commonl}"  used, — and  divided  into  feet  and  inches,  or  feet  and 
tenths  of  a  foot.     The  latter  graduation  is  preferable  for  the 
railroad  engineer,  and  the  former  for  the  city  engineer. 

9.  Eleven  Marking-Pins,  12  or  14  inches  long,  one  of  which 
is  made  of  brass,  the  others  of  No.  4  iron  wire  or  No.  6  steel, 
all  pointed  at  one  end  and  formed  into  a  ring  at  the  other,  are 
used  in  chaining. 

10.  Straight  Poles  about  8  feet  long,  shod  at  the  bottom  with 
a  conical  shoe,  point  down,  and  painted  alternately  red   and 
white  in  foot-width  bands,  are  used  to  indicate  the  direction  of 
the  line  which  is  being  measured,  or  the  position  of  points  to  be 
located.f 

SECTION  II. 

A.     CHAINING. 

11.  Two  men  are  required,  a  "  leader  "  and  a  "  follower,"  or 
head  and  hind  chainman.     The  chain  is  first  thrown  out  in  the 
general  direction  of  the  line  which  it  is  desired  to  measure,  and 

*  Steel  tapes  1000  feet  in  length  have  been  frequently  used  for  special 
purposes.     See  Mine  Surveying,  p.  380. 
t  See  Article  383. 


CHAINING.  5 

examined  carefully  to  see  if  there  are  any  kinks  in  it,  or  bends 
in  the  links ;  the  leader  having  the  inarking-pins  in  one  hand 
takes  hold  of  the  forward  end  of  the  chain  with  the  other,  and 
moves  on  as  nearly  as  he  may  judge  in  the  direction  of  the  line ; 
the  follower  places  the  rear  end  of  the  chain  at  the  station 
whence  the  line  is  to  be  measured,  directs  the  leader  by  signals 
as  he  approaches  the  chain's  length  to  get  in  line,  and  then  calls, 
"halt";  then  the  chain  must  be  drawn  taut  and  straight,  and 
the  follower  having  his  end  of  the  chain  precisely  at  the  starting- 
point,  calls  out,  "down";  the  leader  then  thrusts  one  of  the 
iron  marking-pins  into  the  ground  exactly  at  the  end  of  the 
chain  and  calls  out,  "down,"  which  is  the  signal  to  the  follower 
to  advance :  proceeding  as  before  until  the  second  length  of 
chain  is  measured,  which  is  indicated  by  the  follower  coming  to 
the  pin  set  in  the  ground  b^y  the  leader,  when  the  follower  cries, 
"halt,"  and  after  placing  his  end  of  the  chain  at  the  pin,  the 
chain  having  been  drawn  taut  and  straight  as  before,  calls, 
"down" ;  the  leader,  as  before,  leaving  a  pin  to  mark  the  end 
of  the  chain,  repeats,  "down"  ;  the  follower  then  takes  up  the 
pin  first  placed  by  the  leader,  and  moves  on ;  thus  the  party 
proceeds  until  the  end  of  the  line  is  reached,  the  leader  placing 
the  pins  at  his  end  of  the  chain,  and  the  follower  picking  them 
up  at  his  end. 

If  the  line  ends  with  less  than  the  length  of  the  chain,  the 
leader  places  his  end  at  the  point  which  marks  the  extremity  of 
the  line,  calls  out,  "down";  the  follower  then  reads  off  the 
number  of  links  between  the  last  pin  and  the  end  of  the  line. 
The  number  of  whole  chain's  length  of  the  line  is  shown  by  the 
pins  in  the  hands  of  the  follower,  and  the  number  of  links 
counted  off  added  thereto  will  give  the  total  length  in  chains 
and  links. 

12.  Tally.  If  the  line  exceeds  eleven  chains  in  length,  a 
transfer  of  pins  from  the  hind  chainman  to  the  head  chain  man 
is  necessary ;  this  is  called  tallying,  and  is  performed  in  the 
following  manner:  At  the  end  of  the  eleventh  chain,  the  brass 


6  PLANE  SURVEYING. 

pin  —  the  last  pin  left  in  the  hands  of  the  leader  —  is  placed, 
when  he  call  out  "tally";  at  this  signal  the  follower  drops 
his  end  of  the  chain,  advances  to  the  leader,  counts  over  with 
him  the  ten  iron  pins  which  he  has  gathered  up,  and  transfers 
them  to  the  leader,  who  then  withdraws  the  brass  pin,  sets  an 
iron  one  in  its  place,  and  the  measuring  is  continued  as  before.* 
Each  tally  should  be  recorded,  especially  when  chaining  very 
long  distances,  to  avoid  error  in  the  final  count,  t  It  is  obvious 
that  the  total  length  of  the  line  will  be  equal  to  the  chains  and 
links  as  indicated  above,  plus  the  number  of  tens  shown  by  the 
tallies. 

13.  The  surveyor  should  guard  against  error  in  chaining,  by 
frequently  testing  his  chain,  to  see  that  it  is  of  the  proper 
length,  —  if  it  has  been  stretched,  make  a  file  mark  showing  its 
true  length,  —  and  when  in  use,  see  that  it  is  drawn  straight, 
that  the  forward  chainman  sticks  the  pin  in  line  exactly  at  the 
end  of  the  chain,  or  at  the  mark  indicating  its  true  length,  and 
as  nearly  vertical  as  possible  ;  \  and  when  obtaining  the  number 
of  links  at  the  end  of  the  line,  see  that  they  are  not  counted 

*  Some  surveyors  use  only  ten  marking-pins,  and  tally  by  marking  the 
end  of  the  eleventh  chain  with  a  pencil,  the  finger,  or  a  scratch  on  the 
ground,  and  when  the  ten  pins  are  transferred  to  the  leader,  one  of  them 
is  thrust  in  the  place  thus  indicated,  and  the  work  is  continued  as  before. 

t  In  chaining  long  distances  where  there  are  several  tallies,  the  leader 
and  follower  may,  at  each  tally,  change  places,  and  thereby  lessen  the 
liability  to  error  in  the  final  count.  See  Articles  352,  353. 

}  "  It  has  been  found  by  many  trials  with  as  good  men  as  can  generally 
be  obtained,  that  with  two  sets  of  chainmen  instructed  alike  in  the  proper 
manner  of  keeping  their  chain  level  and  straight  on  the  line,  and  of  setting 
the  tally  pins  plumb,  as  well  as  holding  the  ends  of  the  chain  to  them, 
a  difference  has  sometimes  been  made  of  36  links,  and  an  average  differ- 
ence of  15  or  16  links  to  a  mile  in  common  timbered  land."  — Hurt,  "  Gov- 
ernment Surveying,"  p.  35. 

The  surveyor  should  have  laid  down  by  means  of  a  standard  steel  tape 
or  otherwise,  in  a  convenient  place,  and  between  permanent  marks  in  the 
ground  or  on  the  floor  of  a  large  hall,  the  exact  length  of  a  standard  chain 
by  which  he  could  test  his  chain  from  time  to  time. 


CHAINING.  7 

from  the  wrong  end  of  the  chain,  nor  the  wrong  way  from  the 
brass  mark. 

The  pull  on  the  chain,  when  in  use,  has  a  tendency  to  in- 
crease its  length ;  and  moreover,  since  there  are  a  great  number 
of  wearing  surfaces,  if  each  of  these  be  worn  by  an  extremely 
small  amount,  the  chain  will  be  considerably  elongated. 

In  either  the  surveyor's  or  engineer's  chain  there  are  two 
small  links  which  connect  with  the  two  pieces  of  wire  which 
form  the  principal  part  of  what  is  called  the  'link  of  the  chain, 
thus  giving  six  wearing  surfaces  to  every  link ;  therefore,  if 
each  of  these  surfaces  wears  only  .005  of  an  inch,  the  chain 
will  be  increased  in  length  three  inches,  so  that  in  measuring 
only  a  quarter  of  a  mile  with  a  four-pole  chain,  the  error  from 
this  cause  alone  would  be  Jive  feet,*  making  an  error  in  area 
of  about  4.9  acres  in  a  tract  one  mile  square.  This  stretching 
of  the  chain  is  partially  compensated  by  the  difficulty,  and 
often  impracticability,  of  drawing  the  chain  precisely  straight; 
and  so  long  as  the  chain  is  not  elongated  beyond  one-tenth 
or  one-twelfth  of  one  per  cent  of  its  length,  it  may  be  relied 
on  for  accurate  work.f 

The  true  length  of  a  line  which  has  been  measured  by  a  chain 
stretched  beyond  the  standard  length  may  be  found  from  the 
proportion : 

The  length  of  standard  chain  :  the  length  of  chain  used 
: :  the  distance  measured  :  the  true  distance. 

*  Tliis  error,  it  is  perceived,  increases  directly  with  the  number  of  appli- 
cations of  the  chain  :  it  is  called  cumulative.  The  error  arising  from  erro- 
neous setting  of  the  pin  is  termed  compensative,  that  is,  it  is  as  likely  to  be 
additive  as  subtractive,  and  it  is  shown  by  the  Method  of  Least  Squares, 
that  for  this  class  of  errors  the  square  root  of  the  number  of  errors  are 
probably  not  compensated.  If  the  error  in  setting  is  one  inch,  in  chaining 
a  mile  with  a  Gunter's  chain,  the  probable  error  would  be  VbO  =  about 
9  inches. 

t  To  remove  the  difficulty  of  drawing  the  chain  perfectly  straight,  the 
instructions  issued  from  the  United  States  Land  Office,  1880,  to  Govern 
ment  Surveyors-General,  states  that  the  66  feet  chain  must  be  66.06  feet. 
See  p.  301. 


8  PLANE   SURVEYING. 

For  example,  if,  with  a  chain  stretched  one  link  over  the 
standard,  a  line  be  measured  for  2000  feet,  we  should  have 

100  :  101  =  2000 :  2020,  the  true  distance. 

In  like  manner,  for  the  area  of  a  tract  measured  with  a 
stretched  chain : 

The  square  of  the  length  of  the  standard  chain 
:  the  square  of  the  length  of  the  chain  used 
: :  the  computed  area 
:  the  true  area. 

If  the  chain  was  stretched  one  link,  as  in  the  above  example, 
and  the  area  computed  therefrom  20  acres,  we  should  have 

1002: 1012=  200  sq.  chs. :  204.02  sq.  chs.  for  the  true  area 
=  |$f  of  the  computed  area,  nearly. 

In  general,  if  A  =  true  area,  Al  =  computed  area,  L  —  length 
of  chain,  and  d-L  =  error  in  its  length  (always  small).  Then 
A :  A,  =  (L  ±  dL)2 :  L\ 

Reducing  and  rejecting  d2  as  inconsiderable,  there  results 
A  =  (l±2d)A1;  or,  the  correction  to  be  applied  to  obtain  the 
true  area  =  2  dA^. 

This  correction  is  additive  when  the  chain  is  too  long,  which 
is  the  usual  case,  and  subtractive  when  the  chain  is  too  short. 

14.  The  surfaces  to  be  measured  are  in  general  uneven  and 
broken,  not  plane  ;  but  however  great  the  inequalities,  the  area 
of  a  tract  is  considered  to  be  that  part  of  the  horizontal  plane 
which  is  intercepted  by  vertical  planes  through  its  boundaries.* 
The  horizontal  distance  is  therefore  required ;  hence,  when  the 

*  A  vertical  line  is  a  line  directed  to  the  centre  of  the  earth,  or  it  is  a 
line  having  a  plummet  freely  suspended  to  it,  and  at  a  state  of  rest ;  a 
plumb  line. 

A  vertical  plane  is  a  plane  embracing  a  vertical  line. 

A  horizontal  line  is  a  line  perpendicular  to  a  vertical  line. 

A  horizontal  plane  is  a  plane  perpendicular  to  a  vertical  line. 


CHAINING.  9 

ground  slopes,  it  is  necessary  to  raise  the  down-hill  end  of  the 
chain.  If  the  slope  is  considerable,  only  a  part  of  the  chain 
should  be  used.  For  example,  to  measure  from  L  down  to  JV, 
the  follower  holds  one  end  of  the  chain  at  L,  while  the  leader, 
stretching  the  other  towards  N,  takes  as  much  of  it  as  he  can 
raise  to  a  horizontal  position  &,  and,  holding  a  plummet  there, 
fixes  the  point  c ;  the  follower,  who  is  now  signalled  to  come  for- 
ward, places  at  c  that  point  in  the  chain  whence  the  plummet 
was  suspended  to  fix  c,  while  the  leader  advances  and,  using 
as  much  of  the  chain  as  possible,  locates  e,  and  so  on :  when 
the  end  of  the  chain  is  reached,  a  pin  should  be  transferred 


from  the  leader  to  the  follower.  Where  great  accuracy  is  not 
required,  a  marking-pin  or  pebble  may  be  dropped  to  indicate 
the  points  c,  e,  etc.*  To  measure  up  hill  from  JVto  L  is  less 
accurate,  on  account  of  the  difficulty  experienced  by  the  fol- 
lower in  holding  his  end  of  the  chain  at  the  points  A,/,  d,  etc., 
over  their  counterparts,  i,  g,  e,  etc. 

When  chaining  steep  hills,  especially  if  through  a  wood  or 
over  rough,  rocky  ground,  the  work  may  be  greatly  facilitated 
by  an  extra  chain  man.  He  may  assist  in  getting  line,  straighten- 
ing the  chain,  noting  the  points  c,  e,  etc.,  marked  by  the  plumb 
bob,  and  other  duties. f 

*  If  in  connection  with  the  chain  a  survey  is  being  made  with  an 
instrument  for  measuring  angles,  —  vertical  and  horizontal,  —  the  inclina- 
tion of  a  slope  may  be  observed,  and  the  length  of  it  measured ;  then 
the  horizontal  distance  required  will  be  equal  to  the  measured  distance 
multiplied  by  the  natural  cosine  of  the  angle  of  inclination. 

t   For  extreme  accuracy  in  measuring  lines,  see  Chapter  VII.  Article 


10  PLANE   SURVEYING. 


EXERCISES. 

1.  Set  two  marks  on  gently  undulating  ground  and  about 
1000  feet  apart,  and  measure  forward  and  back  between  these 
points  several  times ;  the  same  party  once  at  least  each  way. 

2.  The  same  between  points  on  hilly  and,  if  possible,  bush 
land. 

3.  Chain  down  a  steep  hill,  and  chain  up  between  the  same 
points. 

B.    RANGING  OUT  LINES. 

15.  If  m  chaining  any  line,  as  LN,  from  L  toward  N,  a  rod 
at  N  can  be  constantly  seen  by  the  rear  chainman,  he  can  keep 

the  leader  in  line  by  ranging  him  with 
L —  — N  the  flagstaff  at  N.  If,  however,  a 

hill  intervenes,  a  valley,  or  brush  or 

woodland  interferring  with  the  alignment,  then  the  line  must  be 
first  ranged  out  or  points  determined  in  it  before  the  chaining 
can  be  performed. 

16.  Ranging  out  a  Line.     To  range  out  a  line  requires  three 
persons,  each  having  a  rod  eight  or  ten  feet  long,  and  a  plum- 
met to  indicate  when  his  rod  is  vertical.     Calling  these  men 
A,  -B,  and  (7,  and  supposing  A  and  B  in  the  line,  C  goes  for- 
ward, and  sighting  back  to  A  and  B,  puts  his  rod  in  line ;  A 
then  advances  beyond  C  and  sets  his  rod  in  line  with  C  and  B ; 
next  B  advances  and  places  his  rod  in  line  with  C  and  A,  and 
so  on  the  line  may  be  extended  any  desired  length.     If,  as 
frequently  is  the  case,  one  of  the  party  has  had  more  experi- 
ence or  is  naturally  better  qualified  for  sighting  a  line,  the  best 
results  would  be  obtained  by  such  an  one  setting  all  the  rods ; 
for  examp\e,  C  would  place  his  rod  in  line,  then  call  up  A,  to 
whom  he  would  turn  over  the  rod  just  set,  and  go  forward  to 
line  the  next ;  after  which  call  up  B,  exchange  rods  with  him, 
and  so  on. 


RANGING   OUT    LINES. 


11 


17.  Over  a  Hill.  To  fix  points  in  a  line  over  a  hill,  both 
ends  of  which  are  visible  from  points  near  the  summit,  proceed 
as  follows : 


.17 


Place  a  flagstaff  at  L,  another  at  N.  A  man  at  E1  signals 
one  at  Z>'.in  line  with  L ;  D'  then  directs  E'  to  E"  in  line  with 
jV;  and  so  on  alternately,  until  the  men  are  at  D  and  E  in  the 
line  LN. 

18.  Across  a  Valley.  To  locate  points  in  a  line,  the  ends 
of  which  may  be  seen  from  each  other,  but  which  are  separated 
by  a  wide,  deep  valley. 


Fix  a  point  C  in  line  with  LN;  then  a  man  holding  a  plumb 
line  at  C,  and  sighting  N  can  direct  the  setting  of  the  stakes 
D,  E,  F,  and  others. 


12  PLANE   SURVEYING. 

19.  Through  a  Wood.  In  chaining  through  a  wood  or  thick 
brush  land,  where  the  ends  cannot  be  seen  from  each  other,  a 
line  *  is  measured  as  nearly  as  may  be  in  the  direction  of  the 
desired  line,  and  stakes  driven  every  two  or  three  chains,  or 
oftener  if  necessary.  When  the  end  of  the  line  is  reached,  the 
distance  to  the  corner  is  measured,  and,  b}*  proportion,  the 
amount  to  move  each  stake  to  bring  it  into  line  is  determined. 


For  example,  let  LNbe  the  true  line,  and  LN1  the  measured 
line  ;  c,  d,  e,  etc.,  points  three  chains  apart.  Now,  if  the  length 
LN'  equals  17.40  chains,  and  NN1  measured  at  right  angles  to 
LN*  =  35  links,  LN-\  will  equal 


and  .  LN' (1740  links)  :NN' (35  links) 

=  Lg    (1500  links)  :  gG    (30  links)  ; 

or  30  links  from  g  at  right  angles  to  LN'  will  indicate  the  posi- 
tion of  G,  a  point  in  the  true  line  LN. 

1740  :  35  =  1200 :  24,  the  distance  fF, 

1740  :  35  =    900 : 18,  the  distance  eE ; 
and  so  on. 

Or,  after  finding  the  first  distance  to  set  off,  either  gG  or  c<7, 
the  others  are  readily  obtained  by  taking  a  proportional  part 
of  this  distance,  shown  by  the  several  divisions  of  the  line  thus  : 
gG  represents  the  fifth  division,  fF  the  fourth,  eE  the  third, 
and  so  on  ;  hence,  if  gG  is  30  links,  fF  will  be  \  of  30,  or  24, 

*  Called  a  random  line  or  trial  line. 

t  If  the  distance  NN'  is  a  s  nail  per  cent  of  the  total  length  of  the  line, 
the  shortest  distance  between  the  ends  of  the  lines  may  be  taken  for  NN', 
and  the  length  of  the  measured  line  for  that  of  the  true  line.  See  Article 
177. 


SETTING   OFF   PERPENDICULARS.  13 

links ;  eE,  f  of  30,  or  18  ;  (W,  f  of  30,  or  12 ;  and  c(7,  |  of 
30,  or  6  links. 

EXERCISES. 

1.  Let  each  student  range  out  a  line  of  several  hundred  feet, 
setting  all  the  poles  forward,  and  hack  again  to  the  starting- 
point,  and  on  different  kinds  of  ground,  undulating,  hilly,  and 
bushy. 

2.  Measure  a  line  through  a  wood  or  where  the  ends  are  not 
visible  from  each  other.     Set  stakes,  as  indicated  in  Article  19, 
in  the  true  line  200  feet  apart.     See  how  near  these  stakes  are 
placed  in  line  by  ranging. 


C.     SETTING   OFF   PERPENDICULARS. 

20.  To  erect  a  perpendicular  at  a  giiven  point  in  a  line. 

Let  MN  be  the  given  line,  and  P  the  point  at  which  it  is 
desired  to  erect  a  perpen- 
dicular. Since  a  triangle 
formed  of  the  sides  3,4,  and 
5,  or  any  multiple  of  these, 
will  contain  a  right  angle, 
we  may  take  parts  of  a  chain 
representing  these  distances  M- 
or  multiples,  having  the  an- 
gle made  by  the  shorter  sides  at  P,  and  set  off  a  perpendicular 
to  a  given  line,  thus:  Fasten  one  end  of  the  chain  at  K,  30 
links  from  P,  the  end  of  the  ninetieth  link  at  P;  then  when 
both  parts  of  the  chain  are  drawn  straight  by  a  pull  at  the 
fiftieth  link,  the  end  of  that  link  will  indicate  the  point  0  which 
if  connected  with  P  will  give  the  perpendicular  required. 

21.  If  the  perpendicular  is  to  be  of  considerable  length,  then 
a  greater  length  than  PO  =  40  links  should  be  used,  and  the 
following  method  would  be  better :  Fasten  one  end  of  the  chain 
at  P,  and  with  the  eightieth  link  describe  an  arc  be ;  measure 


14 


PLANE   SURVEYING. 


PK=  60  links,  and  with  K  as  a  centre,  and  with  a  radius  =  100 
links,  the  whole  length  of  the  chain,  describe  another  arc  de  ; 
the  intersection  of  these  arcs  will  give  the  point  0  required. 

22.  Another  Method.  With  the  whole  length  of  the  chain  as 
a  radius,  and  P  as  a  centre,  describe  an  arc  ab ;  locate  K  a 
chain  from  P,  and  with  the  same  radius,  and  with  a  centre  K^ 

O 


K 


describe  an  arc  cd  cutting  ab  in  Q ;  extend  KQ  to  0,  so  that 
OQ  =  QK,  then  will  OP  be  the  perpendicular  to  the  line  MN 
at  the  point  P.  Why? 


23.    To  let  drop  a  perpendicular  on  a  line  from  a 
without  the  line. 

P' 


!;iven  point 


First,  When  the  point  is  accessible. 

Let  MN  represent  the  line,  and  P  the  point. 


With  a  length 


SETTING   OFF   PERPENDICULARS. 


15 


of  chain  somewhat  greater  than  PO,  describe  an  arc  cutting 
MN  in  the  points  R  and  K.  With  centres  R  and  K,  and  any 
radius  greater  than  the  half  of  RK,  describe  arcs  intersecting 
in  Q.  A  line  drawn  from  P  to  0  in  the  direction  of  Q  will  be 
the  perpendicular  required. 

If  the  point  is  at  P1  at  or  nearly  opposite  one  end  of  the  line, 
extend  the  line  if  it  be  possible  to  N1  until  a  sufficient  distance 
is  obtained  to  describe  the  arcs  required. 

24.  Or  if  it  is  impracticable  to  prolong  the  line,  as  in  the 
figure,  where  a  pond  of  water  prevents,  proceed  as  follows : 


Extend  the  chain  or  any  convenient  portion  of  it  from  P  to 
any  point  R  in  the  line  NO.  Fix  the  middle  point  of  RP,  as 
M,  and  with  this  as  a  centre,  and  a  radius  3fP,  or  its  equal  MR, 
describe  an  arc  cutting  the  given  line  in  0.  Join  PO  for  the 
perpendicular  required.* 

25.    Second,  When  the  point  is  inaccessible. 

Let  P  be  the  given  point,  and  LN  the  line.  At  any  conven- 
ient point  Q  in  the  line  LN  erect  the  perpendiculars  QO  and 
QR  of  equal  length.  Locate  Fin  the  line  PO  and  T  in  the 
line  RP ;  then  if  a  point  S  be  found  at  the  intersection  of  the 

*  The  angle  ROP  is  measured  by  one-half  a  semi-circumference,  and  is 
therefore  a  right  angle. 


16 


PLANE  SURVEYING. 


prolongation  VR  and  OT,  and  a  point  M  be  located  in  ZJVand 
SP,  a  line  joining  M  and  P  will  be  the  perpendicular  sought. 
Why? 


26.  Optical  Square.     To   set   off   perpendiculars    from    a 
line,  an  instrument  called  the  optical  square  may  be  used.     It 
is  a  small  cylindrical  box  containing  a  mirror,  from  the  upper 
half  of  which  the  silvering  is  removed.     The  glass  is  placed  so 
as  to  make  half  a  right  angle  with  the  line  of  sight,  hence  two 
objects  seen  in  it,  the  one  by  direct  vision,  and  the  other  by 
reflection,  subtend  at  the  point  of  observation  a  right  angle. 

Or  the  surveyor's  cross,  which  is  simply  two  pairs  of  sights 
set  at  right  angles  to  each  other,  and  supported  upon  a  staff.* 

D.    RUNNING  PARALLELS. 

27.  Through  a  given  point  to  run  a  parallel  to  a  given  line, 
the  point  and  line  both  being  accessible. 


*  While  these  instruments  may  be  employed  in  chain  surveying, 
neither  of  them  is  used  in  the  ordinary  practice  of  a  surveyor,  as  perpen- 
diculars are  expeditiously  set  off  by  means  of  the  compass  or  transit. 


RUNNING   PARALLELS.  17 

Let  LN  represent  the  line,  and  P  the  point.     Let  drop  a 
perpendicular  PO,  and  at  some  other  point  K\  erect  a  perpen- 


L  0  K  N 

dicular  KR  =  PO.     A  line  drawn  through  P  and  R  will  be  the 
parallel  required. 

28.    Otherwise.    From  any  point  O  in  LN  run  an  oblique 
line  to  the  point  P.     Through  any  point  R  in  PO  measure  a 


N 


Q 


line  MQ,  so  that  RQ  =     ~j~* -—•    A   line  passing  through 

PQ  will  be  the  parallel  required. 

If  R  be  taken  at  the  middle  point  of  OP,  and  QR  be  made 
equal  to  MR,  the  direction  of  the  parallel  PQ  would  be  shown 
at  once. 

B.    OBSTACLES  TO  ALIGNMENT. 

29.    To  prolong  a  line  when  an  obstacle,  as  a  tree  or  building, 

prevents  direct  sighting,  we  may  proceed  as  follows : 


M  PR  S 

By  Perpendiculars.     Let  LN  be  the  line  which  it  is  desired 
to  prolong  past  a  building  B.     At  two  points  0  and  N  in  the 


18  PLANE   SURVEYING. 

line,  set  off  equal  perpendiculars  NP  and  OJf,  of  such  length 
that  a  line  MP  through  these  may  be  extended  past  the 
obstacle  to  some  point  S.  At  R  and  S  set  off  perpendiculars 
to  X  and  F,  of  the  same  length  as  before,  at  0  and  N,  and  join 
XF;  it  will  be  the  prolongation  of  LN. 

30.  Otherwise :  by  Equilateral  Triangles.  On  LN,  the  line 
to  be  prolonged,  take  a  distance  ON  as  a  base,  and  construct 
on  it  an  equilateral  triangle  NOP;  extend  the  side  OP  to  some 


point  Q.  Describe  an  equilateral  triangle  QRS,  and  prolong 
the  side  QR  to  F,  making  QF=  QO;  finally  the  construction 
of  the  equilateral  triangle  VXY  will  give  XT  the  direction 
sought. 

P.    OBSTACLES   TO   MEASUREMENT. 

31.  a.  When  Both  Ends  of  the  Line  are  Accessible. 

By  Perpendiculars.  For  example,  if  it  is  desired  to  measure 
one  side  of  a  field  or  farm  where  a  fence,  hedge,  or  bushes 
prevent  chaining  on  the  line,  set  off  perpendiculars,  and  measure 
the  parallel  line. 

Let  LN  represent  a  line  which,  on  account  of  fence  and  brush, 
it  is  impracticable  to  make  the  measurement  exactly  on  the  line 


OBSTACLES  TO  MEASUREMENT. 


19 


Erect  at  Z,  and  N  perpendiculars  LI  and  Jtfw,  of  equal  and  suffi- 
cient length  so.  that  a  line  connecting  I  and  n  will  clear  the 
obstruction.  Measure  In  ;  it  will  be  the  length  of  the  required 
line. 

32.   b.   When  One  End  is  Inaccessible. 

By  Symmetrical  Triangles.  Suppose  LP  the  line,  P  the 
inaccessible  end,  visible,  but  on  the  opposite  bank  of  a  river. 
Measure  from  any 
point  JVnear  the  river, 
in  a  direction  diverg- 
ing from  its  bank  to  -r 
R,  making  NI=IR. 
Through  any  other 
point  Jf,  in  the  line 
LN,  measure  through 
/  to  K,  so  MI=  IK. 
If  now  a  point  0  be 
found  in  the  prolonga- 
tion of  RK,  and  in 
line  with  /  and  P,  RO  may  be  measured  and  taken  for  their 
distance  NP.* 

33-    Otherwise.      Measure  from  the  line  the   perpendicular 
LP;    erect   at   P  a   perpendicular  to 
PN,  and  extend  it  to  a  point  M  in  the 
prolongation   of  LN.      Measure 
then  the  proportion 

ML:LP=LP:LN 
PL" 
ML 


gives 


LN: 


"  The  student  will  show  that  ROI  and  NIP  are  symmetrical  triangles, 
and  NP  and  RO  are  homologous. 


20 


PLANE   SURVEYING. 


34.  c.    When  Both  Ends  are  Inaccessible. 

By  Symmetrical  Triangles.  Let  LN  be  the  line,  the  length  of 
which  it  is  required  to  determine.  Take  any  point  P,  measure 
PO  and  PM,  and  find  by  one  of  the  preceding  methods  OL, 


MN,  and  hence,  the  total  length  of  PL  and  PN.  Now  take 
points  R  and  Q  in  the  lines  PL  and  PN  respectively,  so  that 
PR :  PQ  =  PL :  PN,  and  measure  RQ ;  then  the  required  line 
LN  may  be  calculated  by  the  proportion  PQ :  PN=  RQ :  LN. 


G.    MEASUREMENT  OF  HEIGHTS. 

35.  To  measure  the  height  of  a  tree  or  a  flag-staff.  Let  BC 
represent  the  height  required.  At  a  point  D  set  up  a  staff  of 
a  known  height  so  that,  with  the  eye  at  -4, 
C  and  E  will  be  in  line  of  sight ;  measure 
AD  and  DB;  then  the  similar  triangles 
ADE  and  ABC  give  the  proportion 

AD  :  DE  =  AB :  BC. 

nn_DExAB 
AD 


Whence 


EXERCISES.  21 

EXAMPLES. 

1.  If  the  height  of  a  staff  is  4  feet,  and  the  distance  from  it 
to  a  tree  =  80  feet,  AD  being  4^  feet,  what  is  the  height  of 
the  tree?  Ans.  77||  feet. 

QUERIES.  If  the  height  of  the  staff  is  equal  to  AD,  the 
length  of  neither  being  known,  simply  the  distance  AB  given, 
could  the  height  of  the  tree  be  ascertained  ? 

If  the  ratio  of  the  height  of  the  staff  to  AD  is  known,  but 
not  the  absolute  length,  could  the  required  height  be  found  by 
simply  measuring  AB1 

Is  this  method  applicable  on  other  than  horizontal  ground  ? 

2.  A  liberty  pole,  whose  height  was  90  feet,  standing  on  a 
horizontal  plane,  was  broken  off,  and  the  extremity  of  the  top 
struck  the  ground  28  feet  from  the  bottom  of  the  pole.     Re- 
quired the  length  of  the  broken  part. 

EXERCISES. 

1.  Set  a  stake  40  feet  perpendicularly  distant  from  a  given 
point  in  a  given  line. 

2.  Through  a  given  point  50  feet  from  a  given  line  run  a 
parallel  120  feet  in  length. 

3.  Prolong  a  line  beyond  a  house  or  other  obstacle. 

4.  Measure  the  width  of  a  stream  or  pond  without  crossing  it. 

5.  Run  a  line  to  the  bank  of  a  stream  or  lake,  and  let  fall  a 
perpendicular  on  the  line  near  its  extremity  from  a  given  point 
without  it. 

6.  Measure  the  height  of  a  tree,  flagstaff,  or  church  spire. 


22  PLANE   SURVEYING. 

SECTION   III. 

RECORDING  THE  FIELD  NOTES. 

36.  The  Field  Notes  should  be  kept  in  a  neat,  concise,  and 
intelligible  manner,  exhibiting  a  complete  record  of  the  work 
done,  and  the  method  of  doing  it,  so  that  a  surveyor  unac- 
quainted with  the  work,  and  having  the  record  before  him, 
could  make  a  plot  of  the  tract,  or  go  on  the  field  and  readily 
ascertain  the  position  of  any  point  indicated  in  the  notes. 

Either  of  two  methods  may  be  employed,  or  a  combination 
of  them. 

37.  Sketch.     One  is  to  make  a  sketch  of  the  tract  as  the 
survey  progresses,  writing  the  length  of  each  line  and  indicating 
the  intersection  of  fences,  roads,  streams,  etc.,  as  shown  below. 


For  surveying  a  field  or  small  tract  of  land,  this  is  a  good 
method,  but  if  the  tract  is  large,  many  sided,  and  numerous 
points  to  be  noted  in  and  'near  the  side-lines  and  diagonals,  it 
would  be  difficult  if  not  impossible  to  decipher  the  sketch  on  a 
page  of  the  ordinary  field-book,  and  to  make  an  intelligible 
record  of  the  work  would  require  a  book  or  sheet  inconveniently 
large  to  carry  about  the  field. 


RECORDING   THE   FIELD   NOTES. 


23 


38.  Columns.  A  method  which  will  answer  as  well  for  com- 
plex as  for  simple  surveys  consists  in  drawing  two  parallel  lines, 
about  an  inch  apart,  extending  from  top  to  bottom  of  the  note- 
book, and  near  the  middle  of  the  left-hand  page.  Between  the 
lines  the  distances  and  stations  are  to  be  recorded,  commencing 
at  the  bottom  of  the  page  and  proceeding  upwards.  Roads, 
fences,  streams,  etc.,  should  be  represented  on  either  or  both 
skies  of  the  column  as  they  naturally  appear.  The  record  of 
the  measurements  on  any  line  being  referred  to  the  beginning 
of  the  line. 

The  right-hand  page  may  be  used  for  sketching  any  part  of 
the  survey  to  further  elucidate,  where  necessary,  the  work  done. 

A  station  is  indicated  by  a  triangle  (A)  or  a  circle  (O) .  If  the 
station  is  at  the  end  of  a  line  it  is  usual  to  name  it  by  the  letter 
or  number,  designating  that  corner  as  station  A  or  station  1,  and 
the  line  extending  from  A  to  B  is  called  the  line  AB^  from  4  to 
5,  the  line  4,  5  ;  or  a  line  may  be  designated 
by  its  length ;  a  line  that  is  3  chains  and 
52  links  long  would  be  referred  to  as  the 
line  352. 


F.  S. 


4.78 


4.78 

A  false  station  is  a  point  in  a  line 
whence  other  measurements  are  to  be  made 
either  to  the  right  or  left,  and  are  desig- 
nated bv  enclosing  in  a  curve  its  distance 
from  the  end  of  the  measured  line,  or  by 
writing  F.  S.  opposite  that  distance,  as 
per  margin,  which  shows  that  there  is  a 
false  station  at  a  distance  of  3.62  chains 
from  A  on  the  line  AB. 

A  fence,  brook,  road,  etc.,  intersecting 
the  measured  line,  should  be  drawn  so  as 
to  indicate,  as  nearly  as  possible,  its  inclination  thereto,  but  not 
as  a  continuous  line ;  the  ends  on  each  side  being  directly 
opposite,  as  at  4.58  and  5.26,  so  that  if  the  vertical  column 


24 


PLANE   SURVEYING. 
O 


4.58 


5.36 


6.18 


were  to  vanish  by  the  two  lines  MN  and  OP  coinciding,  the 
fence  or  creek  would  be  shown  as  continuous. 

When  the  record  of  a  line,  as 
JOT,  is  complete,  and  the  meas- 
urement is  continued  from  N,  a 
horizontal  line  is  drawn  across  the 
column  as  shown  in  the  figure. 
But  if  the  survey  closes  at  the 
end  of  a  line,  as  at  0,  or  if  for 
any  reason  the  work  is  to  proceed 
from  some  other  point,  two  lines 
are  drawn  across  the  column. 


3.40 


2.30 


M 


6.80 


Y 


A  mark  (K)  or  (l~)  placed  at 
the  beginning  of  a  line  indicates 
by  shape,  as  well  as  position, 
that  the  line  along  which  it  stands 
bears  to  the  right  of  the  pre- 
ceding ;  the  reverse  position  of 
the  angle  (N  or  ~l)  indicates  a  turn 
to  the  left. 

In  the  figure,  MN  bears  to  the 
right  of  KM,  and  NO  to  the  left 
QtMN. 


RECORDING   THE   FIELD   NOTES. 


26 


The  record  of  the  survey  sketched  in  Article  37  would  be 
represented  by  the  column  method  as  follows : 


8.35 


9.50 
D 


D 
13.00 

C 


c 

11.25 

5.75 

B 
B 

12.50 

e.oo 

A 


D 
13.50 

IS 

A 

s 

A 

15.00 

i 

0 

26  PLANE   SURVEYING. 

SECTION  IV. 

MAPPING  AND   PLOTTING. 

39.  A  Map  of  a  survey  is  a  correct  representation  or  copy 
of  the  tract  surveyed,  exhibiting  not  only  its  boundaries,  roads, 
streams,  etc.,  in  relative  dimensions  and  positions,  but  also  the 
irregularities  and  appearances  of  its  surface. 

A  Plot  (or  Pkc)  is  an  outline  map,  in  which,  in  general,  only 
the  boundaries,  roads,  streams,  and  important  lines  are  delin- 
eated, but  no  attempt  is  made  to  indicate  the  topography  of  the 
tract.  The  surveyor  usually  makes  a  plot  of  a  field  or  farm 
survey.  The  civil  engineer  makes  a  map  of  a  proposed  railroad. 

INSTRUMENTS  USEFUL  FOR  MAKING  A  PLOT  OF  A  CHAIN  SURVEY. 

40.  Drawing-Board,  T-Square,  Triangles,  Dividers,  Scale, 
Drawing  Pen  and  Pencil.* 

A  Drawing-Board  is  a  rectangular,  smooth  board  to  which 
the  paper  that  is  to  contain  the  drawing  is  fastened.  There 
are  two  patterns :  one  consists  of  a  frame  of  walnut,  or  other 
hard  wood,  with  a  detachable  centre  of  soft  white  pine.  The 
paper,  which  should  be  somewhat  larger  than  the  detachable 
centre,  being  moistened  and  laid  on  it,  becomes  well  stretched 
when  the  parts  of  the  board  are  buttoned  together  and  the 
paper  dries.  The  other  is  simply  a  rectangular  white  pine 
board  made  of  several  pieces  of  wood  laid  in  different  direc- 
tions to  prevent  warping.  Both  patterns  are  made  of  various 
dimensions. 

41.  A  T -Square,  as  its  name  indicates,  is  a  square  or  ruler 
with  a  cross-piece  or  head  at  one  end,  giving  it  the  appearance 


*  Other  instruments  used   in   drawing  are   described   in   Chapter  II. 
Section  VIII. 


MAPPING   AND   PLOTTING.  27 

of  a  letter  T.  There  are  two  patterns  of  these,  one  with  a  head 
fixed  at  right  angles  to  the  ruler  or  blade;  the  other,  in  addition 
to  the  permanent  head,  has  another  head  attached  to  it  with  a 
clamp  screw,  so  that  by  properly  setting  the  movable  head,  lines 
of  any  desired  inclination  may  be  drawn.  The  blade,  being  long 
and  thin,  should  be  tested  occasionally  by  means  of  a  metallic 
straight  edge  or  another  T-square  to  see  whether  or  not  it  is 
perfectly  straight.  The  correctness  of  the  angles  should  also 
be  tested ;  this  may  be  done  as  indicated  in  the  next  article. 

42.  Triangles  are  made  of  hard  wood,  rubber,  or  metal,  and 
are  either  solid  or  have  an  open  centre.     The  angles  are  usually 
30,  60,  and  90  degrees,  or  45,  45,  and  90  degrees,  and  the 
longest  side  rarely  exceeds  12  inches. 

The  T-square  and  triangles  are  frequently  employed  together 
to  draw  parallels,  perpendiculars,  and  many  of  the  oblique  lines 
of  a  plot.* 

The  sides  of  triangles  should  be  tested  occasionally,  to  see  if 
they  are  straight,  by  placing  them  against  the  edge  of  a  metallic 
straight  edge. 

The  right  angle  may  be  tested  by  placing  one  of  its  sides 
against  a  straight  edge ;  mark  the  direction  of  the  other 
side,  reverse  the  triangle,  but  bring  the  same  side  against  the 
straight  edge,  and  having  the  right  angle  at  the  same  point  as 
before,  mark  the  side  again.  If  the  two  marks  coincide,  the 
angle  is  right ;  otherwise,  it  is  not. 

When  correct,  the  right  angle  of  the  triangle  may  be  used  to 
test  the  correctness  of  the  right  angle  of  the  T-square. 

43.  Dividers  (or  Compasses)  are  made  of  different  sizes  and 
numerous  appendages.     The  surveyor  will  need  at  least  one 
with  a  detachable  leg,  so  that  another  leg,  carrying  a  pen  or 

*  The  results  are  tolerably  accurate  within  the  limits  usually  required 
in  a  farm  survey.  It  may  be  well,  however,  to  caution  the  student  not  to 
rely  too  much  upon  the  accuracy  of  a  point  located  by  means  of  and  near 
the  extremity  of  a  thirty-inch  T-square. 


28 


PLANE   SURVEYING. 


pencil  point,  may  be  inserted  when  necessary.  These,  it  need 
hardly  be  said,  are  used  for  laying  off  lines,  describing  arcs, 
circles,  etc. 

44.  Lead-Pencil.     Fine  quality,  hard,  used  in  outlining  the 
work ;   and   a  Drawing-Pen,  medium  size,  for  inking  in  the 
drawing. 

45.  Scales  are  made  of  box-wood,  metal,  ivory,  or  paper, 
and  are  of  various  kinds.     Triangular  and  diagonal  are  gener- 
ally used  for  plotting  chain  surveys.     The  triangular  scale  for 
engineers  and  surveyors  is  usually  12  inches  long,  and  made  of 
good  box-wood,  each  of  the  six  bevelled  faces  being  graduated 
with  a  single  scale,  viz. :  one  face  contains  10  divisions  to  the 
inch,  one  20,  another  30,  another  40,  one  50,  and  one  60  divis- 
ions ;   and  generally  one  inch  on   each   face  is  subdivided  so 
that  an  extremely  small  fraction  of  an  inch  may  be  set  off  or 
read.     This  is  a  very  convenient  scale  ;  not  only  can  very  small 
divisions  be  readily  transferred  from  it  to  a  drawing,  but  by 
simply  placing  the  instrument  properly  on  a  line  of  a  drawing, 
the  scale  of  which  is  known,  its  length  may  be  directly  deter- 
mined. 

The  Diagonal  Scale  is  usually  six  inches  long,  thin  and  flat, 
divided  transversely  into  6  equal  parts  of  one  inch  each,  and 
longitudinally  into  ten  equal  parts.  At  one  end,  as  AD,  one 
inch  is  divided  by  10  oblique  lines,  as  8  wi,  6  ?i,  etc.,  into  10 
equal  parts  and  numbered  as  shown  in  the  figure. 


P  nmA 

Now  Fs  being  .1,  the  next  division  between  the  perpendicular 
FE  and  the  oblique  line  sE  is  .09,  the  next  .08,  and  the  last 


MAPPING    AND   PLOTTING.  29 

division,  or  one  nearest  -F,  is  .01.  Hence  the  scale  may  be 
used  to  measure  .01  of  an  inch,  or  one  hundredth  of  any  divis- 
ion taken  as  the  unit.  For  example,  to  lay  off  3.4,  place 
one  foot  of  the  dividers  at  3  on  the  line  EC  and  extend  the 
other  foot  to  4  between  DE.  To  lay  off  3.42,  place  one  foot 
at  the  intersection  of  3,  3,  and  2,  2,  and  the  other  on  the  same 
line  2,  2,  at  its  intersection  with  4  p. 

The  diagonal  scale  usually  found  with  a  box  of  drawing  in- 
struments contains  various  graduations.  The  simplest  are 
divided  to  inches,  and  halves,  quarters,  tenths,  and  twelfths 
of  an  inch ;  each  quarter  and  half  subdivided  diagonally  into 
tenths,  so  that  a  tenth  of  a  quarter  can  be  taken  off  at  once ; 
and  even  tenths  of  these  are  indicated  on  the  scale  —  besides 
other  divisions  of  more  or  less  utility. 

Paper  scales  are  frequently  employed,  and  regarding  hygro- 
metric  changes  are  better  than  the  others,  for  the  scale  and  the 
paper  containing  the  drawing  expand  and  contract  more  nearly 
alike.  Generally,  however,  they  are  not  divided  with  the  same 
degree  of  accuracy. 

46.  Drawing  to  a  Scale  consists  in  drawing  lines  whose 
length  shall  be  some  fraction  of  the  length  of  the  line  measured. 
Suppose,  for  example,  a  line  is  13  chains  long,  and  it  is  desired 
to  draw  it  to  a  scale  of  5  chains  to  an  inch ;  then  2y6^  inches 
will  evidently  be  the  distance  to  transfer  from  the  scale  to  the 
paper  to  represent  the  length  of  the  line. 

A  line  10  chains  and  50  links  in  length  drawn  to  a  scale  of  3 
chains  to  an  inch  will  be  represented  by  a  line  3|  inches  long, 
and  so  on.  The  length  of  the  line  divided  by  the  number  of 
units  —  chains,  yards,  feet,  etc.  —  to  the  inch,  always  giving 
the  distance  to  be  taken  off  the  scale.  Obviously  the  converse 
of  this  is  true ;  that  is,  the  real  length  of  a  line  may  be  ascer- 
tained when  the  scale  is  known,  by  multiplying  the  units  in  the 
length  of  the  line  in  the  drawing  bv  the  number  of  chains  or 
feet  which  each  unit  represents.  In  the  last  example  the  length 
of  the  line  being  found  3.jr  inches,  and  the  scale  3  chains  to  an 


30  PLANE    SURVEYING. 

inch,  the  true  length  =  3.5  x  3  =  10.50  chains.  The  scale 
should  always  be  given  on  the  drawing.  It  may  be  stated 
thus:  Scale,  3  chains  to  an  inch,  1000  feet  to  an  inch,  2  miles 
to  an  inch,  or  fractionally,  and  thereby  indicating  the  relative 
length  of  the  lines  in  the  drawing  to  those  which  they  repre- 
sent ;  as,  1 :  500,  1 :  2000,  etc. 

47.  Size  of  Drawing  or  Scale  to  Adopt.     In   farm  surveys 
of   small  extent,  1  or  2  chains  to  an  inch  may  be  used ;   for 
medium  tracts  3  chains  to  an  inch  (1  :  2376)   is  perhaps  the 
best.     The  shape  of  the  farm,  the  length  of  the  shortest  and 
longest  sides,  as  well  as  the  object  of  the  drawing,  will,  how- 
ever, influence  the  surveyor  in  his  decision  of  the  scale. 

48.  Scale  Unknown.     If  the  area  of  a  tract  of  land  is  known 
but  the  scale  not  given,  it  ma}'  be  found  by  measuring  the  lines 
of  the  drawing  referred  to  any  convenient  scale  and  comput- 
ing the  area  from  these  determined  lengths.     Then,  since  the 
areas  of  similar  figures  are  to  each  other  as  the  squares  of  their 
homologous  sides,  the  true  scale  may  be  obtained  by  the  pro- 
portion, 

computed  area  _  square  of  assumed  scale  # 
known  area  square  of  true  scale 


SECTION   V. 
ON  AREAS,  AND  ILLUSTRATIVE  EXAMPLES. 

A.     AREAS. 

49.  The  following  are  geometrical  truths  with  which  the 
student  is  supposed  to  have  an  acquaintance,  but  are  given  here 
for  convenience  of  reference. 

*The  protractor  and  other  drawing-instruments  used  in  connection  with 
compass  and  transit  surveying  are  described  in  Chapter  II. 


AREAS.  31 

The  Area  of  a  Triangle  is  equal  to  one-half  the  product  ot  its 
base  and  altitude. 

In  Terms  of  the  Three  Sides  the  area  is  equd  to  the  square  root 
of  the  continued  product  of  one-half  the  sum  of  the  sides,  and 
the  half-sum  minus  each  side  severally,  or  in  symbols,  where 
A.  =  area,  a,  6,  c,  the  three  sides,  and  s  their  sum, 


If  the  triangle  is  equilateral  and  s  =  length  of  a  side, 


50.  The  Area  of  a  Rectangle  is  equal  to  the  product  of  ite 

length  and  breadth,  or  A  =  bl  where  b  =  breadth  and  I  =  length. 

51.  The  Area  of  a  Parallelogram  is  equal  to  the  product  of 
its  base  and  altitude,  or  A='bh  where  &=  breadth  and  h=  height. 

52.  The  Area  of  a  Trapezoid  is  equal  to  the  product  of  one- 
half  the  sum  of  its  parallel  sides  and  the  perpendicular  distance 

between  them,  or  A=^(m+n)  where  m  and  n  are  the  parallel 
sides,  and  p  the  perpendicular  distance  between  them. 

53.  The  Area  of  a  Regular  Hexagon,  where  s  denotes  the 

g 

length  of  one  of  its  sides,  is  ^4  =  -<s2-v/3,  or  it  is  equal  to  six 

equal  equilateral  triangles,  having  for  each  side  the  length  of 
one  side  of  the  hexagon. 

54.  The  Area  of  a  Regular  Octagon,  each  of  its  sides  being 
unity,  may  be  calculated  by  the  rules  of  geometry,  thus  :  Let 
the  figure  represent  the  octagon.     It  is  evident  that  the  area  of 
the  central  square  =  1.      The  sum  of   the  areas  of   the  four 
triangles  m,  n,  o,  _p  =  l,  since   their  sum  equals  the  square 
described  on  db.*     Now,  the  dimensions  of  each  of  the  four 

*  The  square  described  on  the  diagonal  of  a  square  is  double  the  given 
square. 


PLANE   SURVEYING. 


remaining  figures  (rectangles)  #,  y,  z,  and  M,  are  1,  and 
hence  the  sum  of  the  areas  of  these  four  rectangles 

=  4  x  i-V2  =  % V2  ; 
adding  all  the  parts,  there  results 

1  +  1+  2  V2  =  2  +  2  V2 
for  the  area  of  the  octagon. 

55.  The  Area  of  a  Regular  Polygon  in  terms  of  the  perimeter 
and  apothem,  or  radius  of  inscribed  circle,  is  equal  to  one-half 

the  product  of  the  perimeter  and  apothem,  or  A  =  ^ ;  p  denoting 
the  perimeter,  and  r  the  radius  of  inscribed  circle  or  apothem. 

56.  The  Area  of  a  Regular  Polygon  in  terms  of  the  number  of 

sides  and  length  of  one  side  may 
be  determined  as  follows :  Let 
r  =  OP  be  the  radius  of  the  in- 
scribed circle  or  apothem,  I  the 
length  of  each  side,  and  n  the 
number  of  sides,  A  the  area,  as 
before ;  then 


AREAS. 


33 


and 


A      nil      ,  180°      nP     ,  180° 

A  =  —  X  -  cot  —  =  —  cot — 

2       2  n          4  n 


If  1=1,  and  w=8,  the  area  of  the  polygon  (octagon)  becomes 
2  cot  22°  30'=  2  +  2V2,  as  before  found. 

57.  By  the  application  of  the  formulas  just  found,  the  fol- 
lowing table  may  be  constructed,  showing  the  apothems  and 
areas  of  some  of  the  regular  polygons,  each  of  whose  sides  is 
unity. 


NAMES. 

SIDES. 

APOTHEMS. 

AREAS. 

Triangle  .  .  .  . 

3 

0.2886732 

0.4330127 

Square  

4 

0.5000000 

1.0000000 

Pentagon  .... 

5 

0.6881910 

1.7204774 

Hexagon  .... 

6 

0.8660254 

2.5980762 

Heptagon    .  .  . 

7 

1.0382601 

3.6339124 

Octagon    .... 

8 

1.2071069 

4.8284271 

Nonagon  .... 

9 

1.3737385 

6.1818242 

Decagon  .... 

10 

1.5388418 

7.6942088 

Hendecagon  .  . 

11 

1.7028439 

9.3656399 

Dodecagon  .  .  . 

12 

1.  -8660252 

11.1961524 

Now,  since  the  areas  of  similar  polygons  are  proportional  to 
the  squares  on  their  homologous  sides,  this  table  may  be  used 
to  find  the  area  of  any  regular  polygon  named  in  the  table, 
whatever  may  be  the  length  of  its  side.  Using  the  notation 
above,  the  principle  just  enunciated  will  be  expressed  as  follows : 

I2 :  area  in  table  =  P :  A,  or  A.  =  P  X  area  in  table. 

That  is,  the  area  of  a  regular  polygon  is  equal  to  the  square 
of  its  side  multiplied  by  the  area  of  a  similar  polygon  each  of 
whose  sides  is  1 . 

EXAMPLE.  The  area  of  a  regular  pentagon,  each  side  being  30, 
=  302x  1.7204774=1548.43. 


58.    The  Area  of  a  Circle  is  equal  to  TT  multiplied  by  the 
square  of  the  radius,  or  one-half  the  product  of  the  circumfer- 


34  PLANE   SURVEYING. 

ence  and  radius.     Let  R  represent  the  radius,  C  the  circum- 
ference, and  A  the  area  ;  then 


The  area  of  a  Quadrant  =  £=-• 

59.  The  Area  of  a  Sextant  =  -  —  -,  and  in  general,  the  area 

6 

of  any  sector  of  a  circle  =  -^-  X  irR2,  in  which  n  denotes  the 
360 

number  of  degrees  in  the  sector,  or  A  =  -^-,  in  which  I  denotes 
the  length  of  the  arc. 

60.  The  Area  of  a  Circular  Ring  is  evidently  the  difference 
of  the  areas  of  the  outer  and  inner  circles  ;  or,  in  symbols,  if  R 
and  r  equal  the  outer  and  inner  radii,  A  =  -^(R2  —  r2)  . 

61.  The  Area  of  a  Segment  of  a  circle,  as  ABC,  is  evidently 
equal  to  the  area  of  the  sector  AOBC,  minus  the  area  of  the 

C 


triangle  AOB;  or,  in  symbols,  since  the  area  of  the  triangle 

R2  sin  n         ,  ,  , 
=  —  -  -  ,  and  the  area  of  the  sector  as  given  above, 


.  _    Tr 
~~~360~  2 


ILLUSTRATIVE   EXAMPLES.  35 

If  n  is  greater  than  180°,  as  in  the  segment  A'B'BCA,  sin  n 
becomes  negative,  thereby  making  the  second  term  of  the  right- 
hand  member  positive,  as  it  should  ;  since  in  this  case  the 
segment  is  greater  than  the  sector,  and  the  triangle  A'OB'  is 
additive. 

If  the  lengths  of  arc  and  chord  are  given,  denote  by  2c  the 
length  of  chord,  the  other  notation  as  above  ;  then 


the  minus  sign  to  be  used  when  the  segment  is  less  than  a  semi- 
circle, and  the  plus  sign  when  the  segment  is  greater  than  a 
semicircle. 

62.  The  Area  of  an  Ellipse  is  equal  to  irAB,  in  which  A  and 
B  denote  the  semi-axes. 

B.    ILLUSTRATIVE   EXAMPLES. 

EXHIBITING  VARIOUS   METHODS  EMPLOYED  TO  SURVEY  LAND,  TO  PLOT 
THE  SURVEY,  AND  TO  CALCULATE  THE  AREA. 

TRIANGLES. 

63.  First  Method.    Measure  the  perpendicular  (7Z),  and  the 
segments   AD   and  DB,   into 

which  it  divides  the  base  ;  then 
A  _  AB  x  DC 
2 

To  Make  the  Plot.   Draw  AB  A. 

according    to   any  convenient 

scale,  and  locate  D  ;  with  the  same  scale  erect  at  D  a  perpen- 

dicular =  DC.     Join  CA  and  CB,  and  the  triangle  ABC  will 

result. 

tXAMPLES. 

1.  Required  the  area  and  plot  of  a  triangular  field,  the  per- 
pendicular of  which  measures  4.86  chains,  and  divides  the  side 
on  which  it  falls  into  segments  measuring  5.80  chains  and  3.16 
chains,  or  a  total  length  of  8.96  chains. 


86  PLANE   SURVEYING. 

Calculation.  A  =  8-96*4-86  =  21 . 7728  square  chains.  Divid- 
ing by  10,  since  there  are  10  square  chains  in  an  acre,  their 

91   7798 
results  =  2.177  +  acres.*     (The  student  will  make  the 

plot.) 

QUERIES.  Could  a  correct  plot  of  the  tract  be  made  if  there 
were  given  simply  the  base  and  altitude  ? 

Would  there  be,  usually,  any  choice  of  side  to  take  as  the 
base? 

2.  A  triangular  field  measures  12.18  chains  on  one  side,  and 
the  perpendicular  erected  at  a  point  5.10  chains  from  one  end 
measures  7.54  chains.  Calculate  the  area  and  make  the  plot. 

64.  Second  Method.^  Measure  all  the  sides,  and  calculate 
the  area  by  the  formula  given  above  for  that  case. 


EXAMPLES. 

1.  The  lengths  of  the  sides  of  a  triangle  are  as  follows: 
AB=40  chains,  AC =30  chains,  and  .8(7=20  chains.  Re- 
quired the  area  and  plot- 


A  =  V45  x  5  x  15  x  25  =  29.047  acres. 

To  Make  the  Plot.  Take  40  chains  to  any  convenient  scale 
in  the  dividers,  and  lay  it  off  for  the  base  AB;  then,  with  A 
as  a  centre  and  30  chains  to  the  same 
scale  in  the  dividers,  describe  an  arc 
mm' ;  also,  with  B  as  a  centre  and  20 
chains  for  radius,  describe  the  arc  nn'. 

_  _  The  point  C  connected  with  A  and 

B  will  give  the  triangle  ABC  required. 
REMARK.     It  is  customary  when  making  a  chain  survey,  to 

*  The  area  is  usually  expressed  in  acres  and  hundredths  or  thousandths 
of  an  acre. 

t  Other  methods  are  given  in  Chapter  II.  Section  IX. 


ILLUSTRATIVE  EXAMPLES.  37 

measure  a  proof*  line  such  as   CD,  and  this  should  always 
be  constructed  to  test  the  accuracy  of  the  work. 

2.  The   three   sides   of   a   triangle  measure  49,  50.25,  and 
25.69  chains.     Find  the  area.  Ans.    61.498  acres. 

3.  The  sides  of  a  triangular  field  are  24,  18,  and  15  chains. 
A  proof  line,  12  chains  in  length,  intersects  the  longest  side  or 
base  at  a  point  10.25  chains  from  the  angle  formed  by  the 
two  longest  sides  of  the  field.      Required  the  area  and  plot. 
Test  accuracy  of  latter  by  constructing  proof  line. 


RECTANGLES. 

65.    Measure  any  two  adjacent  sides,  as  AB  and  BC.     The 
area  =  A  =  AB  •  BC.  D 

To  Plot.  Lay  off  AB  to  any  de- 
sired scale,  and  erect  a  perpendicular 
with  the  same  scale  at  the  extremities 
=  AD  and  BC ;  connect  D  and  (7, 
and  the  required  figure  will  be  formed.  A. 

EXAMPLES. 

1.  The  length  and  breadth  of  a  rectangle  are  12.32  and  7.16 
chains  respectively.     Required  the  area.  Ans.    8.82  acres. 

2.  The  length  of  a  rectangle  is  1250  feet,  and  its  bread th'840 
feet.     What  is  its  area?  Ans.    24.1  acres. 

3.  A  road  running  across  a  farm  is  ^  of  a  mile  long  and  3 
rods  wide.     How  much  laud  does  it  occupy  ?       Ans.    2\  acres. 

4.  The  length  of  a  road  on  a  hillside  inclined  to  the  horizon 
at  an  angle  of  20°  is  2310  feet,  and  its  width  2|  rods.     At  the 
rate  of  $84  per  acre,  what  must  be  paid  to  the  owner  across 
whose  land  the  road  runs?  Ans.    $189.93. 

*  A  line  to  check  the  measurement. 

48604 


38  PLANE   SURVEYING. 

PARALLELOGRAMS. 

66.  Measure  a  side,  as  AB,  the  perpendicular  distance,  as 
j>  E       c  BE,  to  the  opposite  side  DC,  and  the 

/  i     /     distance  CE.     Then  A  =  AB  x  BE. 

/  •/  To  Plot.    La\-  off  the  base  AB,  and 


-4  B  at  the  extremity  B  erect  a  perpendicu- 

lar equal  BE.  Through  E  draw  DC  equal  to  and  parallel  to 
AB,  making  EC  its  proper  length.  Join  DA  and  CB,  and  the 
parallelogram  ABCD  will  be  formed. 

EXAMPLES. 

1.  The  base  of  a  parallelogram  measures  10.54  chains.     A 
perpendicular  from  one  extremity  of  the  base  to  the  opposite 
side  5.16  chains,  and  the  distance  corresponding  to  EC  in  the 
last  figure  is  1.82  chains.     Required  the  area  and  plot. 

Ans.    5.439  acres. 

2.  A  surveyor  employed  to  determine  the  area  of  a  rhombus, 
and  knowing  that  the  obtuse  angles  were  xiouble  the  acute, 
measured   the  shorter  diagonal  only,  and  found  it   100  feet. 
Was  the  measurement  sufficient?     If  so,  give  the  area. 

QUERIES.  Can  the  area  of  a  rhombus  be  ascertained  if  the 
lengths  only  of  the  two  diagonals  be  given  ?  If  either  diagonal 
and  a  side  be  given  ? 

TRAPEZOIDS. 

67.    Measure  EC,  the   perpendicular    CD,    and   BA ;    note 

E C      where   the   perpendicular    CD  meets   the 

/  ~A      base  AB. 


CD. 


A  D  B 

To  Plot.  Lay  off  the  base  AB  to  the  desired  scale,  and 
at  D  erect  a  perpendicular  thereto  equal  to  DC.  Through 
C  draw  CE  of  the  required  length  and  parallel  to  AB.  Join 
EA  and  CB.  The  figure  resulting  will  be  the  trapezoid 
required. 


ILLUSTRATIVE   EXAMPLES.  39 

EXAMPLES. 

1.  The  base  of    a  trapezoid   measures   12.62   chains,   the 
parallel  side  8.14  chains,  and  the  perpendicular  7.44  chains. 
The   distance   corresponding  to  DB  in  the  last  figure  is  1.12 
chains.     Required  the  area  and  plot.  Area=  7.723  acres. 

2.  A  railroad  embankment  extends  3240  feet  perpendicularly 
across  a  farm  intersecting  parallel  sides.     At  one  end  its  base 
is  96   feet  wide,   and  at  the  other  60  feet.      Supposing   the 
property  line  is  10  feet  from  the  embankment  on  each  side, 
how  much  of  the  farm  is  taken  for  railroad  purposes? 


TRAPEZIUMS. 

68.    First  Method.     Measure  either  diagonal,  and  the  per- 
pendiculars thereto  from  the  opposite 
angles,  noting  the  distances  AH  and 
EC. 


To  Plot.     Draw  the  diagonal  AC  to  AL^ ^ 

the  desired  scale,  and  fix  the  points  H 

and  E.  At  these  points  erect  perpendiculars  corresponding  to 
the  scale  and  measurements.  Joining  DA  and  DC,  and  BA 
and  BC,  will  complete  the  plot  required. 

Second  Method.  Measure  all  the  sides 
and  a  diagonal  as  shown  in  the  figure, 
thereby  dividing  the  trapezium  into  two 
triangles,  all  the  sides  of  which  are 
known  ;  whence  the  area  may  be  com- 
puted by  the  formula  for  the  area  of  a 
triungle  in  terms  of  the  three  sides. 

To  Plot.  Lay  off  the  diagonal  AC,  and  locate  the  points  B 
and  D  by  methods  heretofore  given.  Connect  the  points 
ABC  I)  A  for  the  plot  required. 


40  PLANE   SURVEYING. 


EXAMPLES. 

1.  The  diagonal  of  a  trapezium  measures  120  rods,  and  the 
two  perpendiculars  30  and  40  rods ;  what  is  the  area  ? 

Ans.    26^  acres. 

2.  The  sides  of   a   trapezium   taken   in   regular  order   are 
AB  =  5,   BC=9,    CD  =  11,    and    DA=13    chains,    and   the 
diagonal  AC—  12  chains.     Required  the  area  and  plot. 

3.  The  sides  of  a  trapezium  are  18.10,  22.14,  28.16,  and 
34.62  chains,  and  the  diagonal  from  the  first  to  the  third  corner 
is  30.76  chains.     Determine  the  area. 


POLYGONS. 

Regular  or  irregular,  five  or  more  sides. 
69.    First  Method.     Measure  all  the  sides  and  the  diagonals, 
thus  dividing  the  tract  into  three  or  more  triangles.     The  area 
will  equal  the  sum  of  the  areas 
of  the  triangles  thus  formed. 

To  Plot.  Draw  a  line  repre- 
senting the  diagonal  BE,  and 
construct  the  triangle  ABE  on 
it ;  on  the  other  side  of  BE  con- 
struct BCE;  if  a  pentagon,  the 
plot  will  be  completed  by  add- 
ing ODE. 

If  a  hexagon,  there  must  be 
measured  another  diagonal  giving  four  triangles,  and  generally, 
for  any  number  of  sides  n,  there  will  be  n  —  3  diagonals  and 
n  —  2  triangles,  the  area  of  the  tract  being  equal  to  the  sum 
of  the  areas  of  the  n  —  2  triangles. 

If  the  tract  be  a  regular  polygon,  the  measurement  of  one 
side  by  the  aid  of  the  table  in  (57)  will  be  sufficient  to  deter- 
mine the  area. 


ILLUSTRATIVE  EXAMPLES. 


41 


70.  Second  Method.*  Measure  one 
or  more  diagonals,  and  perpendiculars 
from  these  to  the  opposite  angles,  or 
corners,  thereby  dividing  the  tract  into 
right  triangles,  or  right  triangles  and 
trapezoids.  The  sum  of  the  areas  of 
these  figures  will  equal  the  area  of 
the  polygon. 

EXAMPLES. 

1.  The  sides  of  a  pentagon  taken  in  regular  order  are,  6.80, 
4.20,  5.30,  8.90,  and  9.62  chains.      The  diagonals  from  the 
fifth  corner  to  the  second  and  third  are  each  10  chains.     Find 
the  area,t  and  make  a  plot. 

2.  A   side   of    a    regular   heptagon   measures   4.25   chains. 
What  is  the  area? 

Given   the   following  field  notes  to  calculate  the  areas  and 
make  the  plots.     The  distances  are  in  chains. 


D 

D 

16.75 

12.50 

C4.50 

13.50 

7.80 

4.60  E 

12.90 

4.50  E                                C  2.80 

5.90 

9.00 

S'80  F                                 B  5.50 

3.20 

B  4.50 

4.80 

2.60 

3.00  F 

8.20 

6.25  G 

A 

A 

*  Other  methods  are  given  in  Chapter  II. 

t  The  work  may  be  abridged  by  using  logarithms. 


42 


PLANE   SURVEYING. 


•  -Proof  Lines 7  -  -  r  -  -  -> 


5  re 


8    S  £ 


s  s 


ILLUSTRATIVE   EXAMPLES.  43 


CIRCLES  AND  CIRCULAR  RINGS. 

71.    Measure  the  radius  or  diameter  of  a  circle,  and  the  radii 
or  diameters  of  a  circular  ring. 

The  area  of  the  former  =  TT^  =  — . 
The  area  of  the  latter    =  *(!?  —  r2). 

EXAMPLES. 

1.  The  diameter  of  a  circle  is  10.16  chains.     What  is  the 
area? 

2.  What  is  the  'area  of  a  circular  ring,  the  outer  and  inner 
radii  measuring  respectively  20  and  12  rods? 


SECTORS  AND  SEGMENTS. 

72.  Measure  the  chord  AB,  and  the  perpendicular  distance 
or  height  of  arc  DE,  from  the  centre  of 
AB  to  the  arc  AEB.  From  these  data 
the  radius  and  the  angle  at  the  centre 
may  be  found ;  and  hence  the  area  ob- 
tained. See  (59)  and  (61).  Otherwise, 
measure  the  radius  BC,  and  by  short 
chords  the  arc  AEB;  whence  the  area 
may  be  computed.  (The  student  will 
supply  the  details  for  both  cases.)  G 

EXAMPLES. 

1.  If  the  length  of  the  arc  of  a  sector  is  500  feet  and  the 
radius  1000  feet,  how  many  acres  does  it  contain?  Ans.  5.739. 

2.  If  the  chord  AB  (last  figure)  =40  feet,  and  the  height  of 
arc  DE  =  10  feet,  what  is  the  area  of  the  segment? 

Ans.   279.558  square  feet. 

3.  Given  the  radius,  which  is  bisected  by  the  chord,  =  100 
feet.     Required  the  area  of  sector  and  segment. 


44 


PLANE   SURVEYING. 


SECTION  VI. 

OFFSETS  AND  TIB-LINES. 

73.  When  any  portion  of  the  boundary  of  a  tract  of  land  is 
irregular,  as,  for  example,  when  it  is  a  stream  or  crooked  road, 
the  survey  along  such  sides  is  best  effected  by 
measuring  a  straight  line,  as  LN,  and  setting 
off  short  perpendiculars  ra'ra,  o'o,  and  p'p  at 
points  m',  o',  and  p',  and  extending  them  to 
the  boundary  line.  Such  short  perpendiculars 
are  called  offsets,  and  they  should  be  so  chosen 
,0  that  the  part  of  the  curve  Lm,  mo,  op,  etc., 
/m  intercepted  between  any  two  consecutive  ones 
may  be  considered  straight ;  whence  the  area 
of  the  part  lying  between  the  straight  and 
curved  lines  may  be  obtained  by  adding  to- 
gether the  area  of  the  triangles  and  trapezoids  into  which  it  is 
thus  divided. 

If  the  field  notes  corresponding  to  the  above  figure  are  as 
below : 


N 
9.00 
7.00 
2.50 
1.20 

L 


0 

1.30  p 
1.40  o 
1.00  m 
0 


The  area  between  straight  line  and  boundary 

f  Area  triangle        Lmm',      6000  square  links, 

I  Area  trapezoid    mm'oo',     15600          "  " 

"~  1  Area  trapezoid       oo'pp',     60750          "  " 

[Area  triangle          p'pN,     16900          "  " 

Their  sum  =  99250          "  " 

cr,  .9925  of  an  acre. 


OFFSETS   AND   TIE-LINES. 


45 


x' 


74.  Rectangular  Co-ordinates.     Let  XX'  and  YT'  be  two 

straight  lines  intersecting  each  other  at  right  angles  at  0,  and 
P'P",  points  in  their  plane. 
Then  if  perpendiculars  be  drawn 
thi-ough  these  points  to  the  lines 
XX'  and  YY',  the  distances 
cut  off  on  the  former  are  called 
abscissas,  and  those  on  the  lat- 
ter ordinates.  The  abscissa  and  ^ 
ordinate  referring  to  one  point, 
as  P,  are  termed  the  co-ordi- 
nates of  that  point.  , 

The  lines  to  which  the  meas- 
urements are  referred  are  called  the  axes;  XX'  being  called 
the  axis  of  abscissas  or  axis  of  X,  and  YY'  the  axis  of  ordi- 
nates or  axis  of  Y. 

The  axes  being  at  right  angles,  the  system  is  called  the 
rectangular  system  of  co-ordinates.  0  is  the  origin.*  Desig- 
nating the  ordinates  measured  from  the  axis  of  X  upward,  and 
the  abscissas  measured  to  the  right  of  the  axis  of  Y,  as  plus, 
and  those  downward  from  the  X-axis  and  to  the  left  of  the 
F-axis,  as  minus,  it  is  evident  that  a  point  can  be  located  in 
either  quadrant  very  readily  by  this  method. 

If  the  co-ordinates  of  P1  are  x  =  6  and  y  =  4,  it  means 
simply  that  Ox'  =  6  and  Oy'  =  4,  and  the  point  may  be  located 
by  drawing  the  lines  as  indicated.  If  x  =  —  5  and  y=3,  the 
point  is  five  units  to  the  left  of  the  F-axis,  and  three  units 
above  the  X-axis,  etc. 

75.  Application  of  Rectangular  Co-ordinates  to  the  Computa- 
tion of  Areas. 

Suppose  it  is  required  to  find  the  area  of  any  number  of 
trapezoids  formed  by  a  broken  line,  and  perpendiculars  from  its 
angles  upon  a  straight  line  as  indicated  in  the  figure.  XX',  the 


*  Axes  inclined  to  each  other  are  called  oblique. 


46 


PLANE   SURVEYING. 


straight  line,  may  be  taken  as  the  axis  of  X,  and  TV  the  axis 
of  T.  Let  #„  xm,  xn,  etc.,  y»  ym,  yn,  etc.,  denote  respectively 
the  abscissas  and  ordinates  of  the  points  L,  M,  N,  etc. 


I'm 


Yn 


F, 


XI  Xm 


The  area  required 


By  expanding  and  simplifying  there  results 

(ym  -  y» 


Whence  for  calculating  the  area  of  a  tract  of  land  included 
between  a  straight  line  and  a  broken  line,  whose  angles  are 
given  by  their  co-ordinates  upon  the  straight  line  as  base,  we 
have  the  following 

RULE. 

Multiply  the  difference  between  each  ordinate  and  the  second 
succeeding  one  by  the  abscissa  of  the  intervening  ordinate. 

Multiply  also  the  sum  of  the  last  two  ordinates  by  the  last 
abscissa. 

The  half  of  the  algebraic  sum  of  these  several  products  will  be 
the  area. 

EXAMPLES. 

Calculate  the  areas,  and  make  the  plots  from  the  following 
field  notes  ;  the  distances  are  in  chains. 

*  A  similar  expression  could  evidently  be  found  for  any  number  of 
trapezoids. 


OFFSETS   AND   TIE-LINES. 


47 


13.60 

.90 

12.40 

1.50 

9.80 

2.10 

5.80 

1.60 

2.80 

1.00 

1.00 

.60 

0 

0 

18.90 

1.60 

16.70 

2.00 

12.40 

2.40 

7.40 

1.50 

4.20 

1.00 

1.70 

.20 

0 

0 

76.  A  slight  modification  of  the  rule  just  given  will  make  it 
applicable  to  the  case  where  a  broken  line  encloses  a  tract  or 
forms  the  boundary  of  a  polygon. 


Let  the  tract  enclosed  be  represented  by  the  figures,  then 
the  area 


48  PLANE   SURVEYING. 


By  expanding,   cancelling,    and    factoring,  we  may   obtain 
either  of  the  following  expressions  : 

(y0  -  &,) 


(a;,  -  x.)  ].-         (2) 

Whence,  for  the  area  of  a  polygon  whose  corners  are  given 
by  their  co-ordinates,  we  have  the  following 

RULE. 

Take  one-half  the  sum  of  the  products  of  each  •{ 
and  the  difference  of  its  adjacent  <{  ^^^^  <l 
the  subtraction  in  the  same  direction  round  the  plot.* 

EXAMPLES. 

1.  Given  the  abscissas  of  the  several  corners  of  a  field,  L, 
M,  N,  0,  P,  respectively  : 

2.00,  5.50,  12.00,  15.00,  and  8.60  chains. 
The  corresponding  ordinates  : 

10.20,  1.80,  4.00,  9.40,  and  14.00  chains; 
to  compute  the  area. 

*  The  work  of  computation  may  be  abridged  when  the  abscissas  are 
greater  than  the  ordinates,  by  making  the  differences  of  the  a"bscissas  the 
factors  with  the  ordinates  ;  and  when  the  ordinates  are  greater  than  the 
abscissas,  taking  the  differences  of  the  ordinates  with  the  abscissas.  If 
the  axis  of  ordinates  pass  through  L,  the  abscissa  of  that  point  would 
vanish.  Regard  must,  in  all  cases,  be  had  to  the  resulting  signs. 


OFFSETS   AND   TIE-LINES. 
The  form  of  reduction  is  as  follows : 


49 


CORNERS. 

OBDINATES. 

ABSCISSAS. 

DIFFERENCE  BETWEEN 
ALTERNATE  ABSCISSAS. 

DOUBLE  AREAS. 

L 

10.20 

2.00 

3.10. 

31.6200 

M 

1.80 

5.50 

-  10.00 

-   18.0000 

N 

4.00 

12.00 

-   9.50 

-  38.0000 

0 

9.40 

15.00 

3.40 

31.9600 

P 

14.00 

8.60 

13.00 

182.0000 

245.5800 

-  56. 

v 

2)189.5800 

10)94.79    sq.  chs. 

9.479  acres. 

2.  Given  the  abscissas  of  the  several  corners  of  a  field,  Lt 
M,  N,  0,  P,  Q,  R,  respectively : 

0,  6.50,  14.60,  22.80,  20.00,  16.70,  9.90; 
and  the  corresponding  ordinates : 

13.20,  3.72,  4.40,  3.90,  17.24,  16.90,  and  17.30, 
all  in  chains ;  to  determine  the  area  and  make  a  plot. 

3.  Given  the  abscissas  of  the  several  corners  of  a  field,  A, 
B,  (7,  D,  E,  F,  G,  H,  respectively : 

100,  300,  360,  290,  400,  250,  120,  0; 
and  the  corresponding  ordinates : 

0,  0,  160,  300,  380,  520,  520,  and  330, 
all  in  feet ;  to  determine  the  area,  and  make  a  plot. 


4.    Verify  Example  3  by  a  method  independent  of  that  given 
on  the  preceding  page. 


50  PLANE   SURVEYING. 

5.    Required  the  area  and  plot  from  the  following  field  notes : 


t 

A 

E 

A 

23.50 

41.10 

y 

F 

Y                             Diag.  DF 

25.80 

F 

§ 

21.90 

Diag.  CF 

17.65 

£ 

E 

Y 

B 

V 

E 

F 

A 

20.50 

30.10 

D 

Y 

D 

A 

D 

C 

*e 

0 

17.40 

28.90 

ID 

1.00 

15.50 

N 

F 

ii 
I 

T3 

1.60 

13.00 

F 

OQ        fi 

5 

1.75 
1.00 
1.20 

8.50 
6.00 
3.50 

26.75 
B 

i 

0 

0 

C 

K 

1 

C 

On  river  bank. 

SB 

0 

18.00 

o 

2.00 

13.50 

3.50 

10.00 

2.50 

5.50 

V 

0 

B 

Y 

j 

B 

! 

18.50 

* 

i 

A 

Other  examples  containing  offsets  are  given  in  Chapter  II, 


OFFSETS   AND   TIE-LINES. 


77.  To  find  the  area  of  a  tract  of  land  when  it  is  impossible 
to  measure  the  diagonals  or  perpendiculars,  as  in  the  case  of  a 
lake  or  swamp. 

Measure  MN  and  ON,  and  continue  the  measurements  past 
their  intersection  at  N,  making  NH 
some  fractional  part  of  MN,  and  NK 
the  same  part  of  ON.*  Now  because 
of  the  similarity  of  the  triangles  MNO 
and  HKN,  MO  may  be  found  by 
measuring  a  tie-line  HK,  and  divid- 
ing it  by  the  fraction  used.  Similarly, 
LM  may  be  found^  Then  OL  being  , 
measured,  the  area  of  the  polygon 
MNOLR  can  be  computed.  In  case 
of  a  pond  or  lake,  if  offsets  be  taken 
from  the  sides  of  the  polygon  to  the 
edge  of  the  water,  and  the  sum  of  the 
areas  thus  found  included  between 
the  sides  and  the  lake  be  deducted 
from  the  area  of  the  polygon,  the  area 
of  the  body  of  water  will  be  shown. 


MISCELLANEOUS    EXAMPLES. 

1.  One  side  of  an  equilateral  triangle  measures  18.24  chains. 
Required  the  area. 

2.  The  perpendicular  of  an  equilateral  triangular  piece  of 
ground  measures  160  feet.     What  is  the  area? 

Ans.    14780.16  square  feet.     What  part  of  an  acre? 


*  Great  care  should  be  exercised  in  the  measurements,  since  the  erroi 
is  magnified  in  the  computed  lines.  If  the  lines  are  so  taken  that  KH  is 
one-fourth  of  MO,  an  error  of  one  link  in  measuring  KH  will  make  a 
difference  of  four  links  in  MO. 

For  methods  of  performing  such  work  more  accurately,  see  Compass 
and  Transit  Surveying,  Chapter  II.  Section  IX. 


62  PLANE   SURVEYING. 

3.  It  is  known  that  the  base  of  an  isosceles  triangle  is  f  the 
length  of  one  of  its  equal  sides.     The  perpendicular  measures 
80  feet.     The  sides  and  area  are  required. 

Ans.   Each  side,  100  feet;  base,  120  feet. 
Area,  4800  square  feet. 

4.  Desiring  to  ascertain  the  radius  of  a  railroad  curve  (it 
being  the  boundary  of  a   field),  a   surveyor   measured   from 
centre  to  centre  of  tracks,  a  chord  of  200  feet ;  also  the  per- 
pendicular distance  from  the  centre  of  chord  to  the  middle  of 
tracks,  4  feet.     Show  that  these   measurements   indicate   the 
radius  =  1252  feet. 

QUERY.  How  should  the  data  obtained  in  Example  4  be  em- 
ployed to  determine  the  area,  assuming  that  the  curve  is  con- 
cave to  the  field  ? 

5.  The  circumference  of  a  circle  is  100  rods.     How  many 
acres  does  it  contain  ?  Ans.    4.974. 

QUERY.  Can  Problem  5  be  solved  without  first  finding  the 
radius  or  diameter? 

6.  If  the  number  expressing  the 'area  of  an  equilateral  tri- 
angle in  square  feet  is  the  same  as  that  showing  the  length  of 
one  of  its  sides  in  lineal  inches,  what  is  its  area? 

Ans.    332.55. 

7.  The  chord  of  a  circle  measures  60  feet,  and  the  height  of 
arc,  or  versed  sine,  10  feet.     Find  in  the  same  circle  the  versed 
sine  of  a  chord  of  90  feet.  Ans.    28.2  feet. 

8.  The  lengths  of  two  chords  lying  on  the  same  side  of  the 
diameter  of  a  circle  are  96  and  60,  and  their  distance  apart  26. 
Required  the  area  between  them. 

SUGGESTION.  Let  x  =  perpendicular  distance  from  centre  of 
short  chord  to  the  nearest  point  of  circumference,  and  y  =  per- 
pendicular distance  from  centre  of  long  chord  to  the  farthest 
point  of  circumference  ;  that  is,  measured  in  the  opposite  direc- 
tion from  the  first 


OFFSETS    AND  TIE-LINES.  58 

Then  *  (y  +  26)  =    900. 

y  (x  +  26)  =  2304. 

Whence  the  diameter  is  readily  determined  and  thence  the 
area  required. 

9.  Show  that  the  area  of  the  circumscribed  hexagon  is  to 
the  area  of  the  circumscribed  equilateral  triangle  as  2  is  to  3. 

10.  Show  that  the  area  of  a  regular  inscribed  polygon  of  n 

. ,         n   ,    .    360° 

sides  =  -  r2  sin 

2  n 

11.  Show  that  the  area  of  a  regular  circumscribed  polygon 

1  &0° 

of  n  sides  =  w2  tan • 

n 

12.  The  distance  between  the  centres  of  two  circles,  whose 
diameters  are  each  50,  is  equal  to  30.    What  is  the  area  common 
to  the  two  circles?  Ans.   559.15. 

13.  Three  equal  circles  being   tangent   to   each   other  ex- 
ternally enclose  40  rods.     What  is  the  radius  of  each  circle? 

Ans.   15.75  rods. 


EXERCISES. 

1.  Survey  a  polygon,  measure  all  the  sides  and  necessary 
diagonals,  run  test-lines,  record  the  notes,  make  a  plot,  and 
compute  the  area. 

2.  Take  the  boundaries  as  found  above,  and  complete  the 
survey  by  measuring  one  diagonal  and  perpendicular  offsets  to 
the  corners.     Make  record,  plot,  and  computation. 

3.  Measure  a  field  partly  bounded  by  a  creek  or  lake,  ren- 
dering it  necessary  to  take  offsets  thereto.     Record  the  notes, 
plot,  and  calculate  area. 

4.  Survey  a  pond  or  small  lake  by  tie-lines  and  offsets. 


CHAPTER  H. 


COMPASS  AND  TEANSIT  SUEVEYING. 


SECTION   I. 

DEFINITIONS  AND  DESCRIPTION  OP  INSTRUMENTS. 

78.  The  Axis  of  the  earth  is  the  imaginary  line  about  which 
it  rotates. 

The  Poles  are  the  points  where  the  axis  pierces  the  earth  :  one 
the  north  pole,  the  other  the  south  pole. 

79.  A  Meridian  Plane  is  a  plane  embracing  the  earth's  axis. 

80.  A  Meridian  Line,  or  true  meridian,  is  the  intersection  of 
a  meridian  plane  with  the  surface  of  the  earth. 

In  plane  surveying  the  meridians  passing  through  the  ex- 
tremities of  lines  surveyed  are  considered  parallel. 

81.  The  Magnetic  Needle  is  a  thin  bar  of  strongly  magnet- 
ized steel,  balanced  on  a  pivot,  so  that  it  may  turn  freely,  and 
always  come  to  rest  in  the  direction  of  the  magnetic  meridian. 

82.  The  Magnetic  Meridian  is  indicated  by  the  direction  of 
a  bar  magnet,  when  horizontal,  freely  suspended  and  at  rest. 
It  does  not  in  general  coincide  with  the  geographic  meridian. 
The  angle  included  between  them  is  called  the  declination  of  the 
needle,  or  variation  of  the  compass.*  and  the  change  in  this 
angle  is  termed  the  variation  of  the  declination. 

*  See  Chapter  III.,  on  Declination  of  the  Needle. 


DESCRIPTION   OF  INSTRUMENTS.  55 

83  The  Azimuth  of  a  Line  is  the  angle  which  the  vertical 
plaue  containing  it  makes  with  the  plane  of  the  meridian. 

84.  The  Bearing  of  a  Line,  called  also  the  course,  is  the 
angle  which  it  forms  with  the  direction  of  the  magnetic  needle. 

85.  The  Meridian  Distance  of  a  Point  is  its  perpendicular 
distance  from  an  assumed  meridian. 

86.  The  Meridian  Distance  of  a  Line  is  the  meridian  dis- 
tance of  the  middle  point  of  that  line. 

87.  A  Horizontal  Angle  is  an  angle  included  between  two 
lines  in  a  horizontal  plane. 

A  Vertical  Angle  is  an  angle  included  between  two  lines  in 
a  vertical  plane. 

88.  An  Angle  of  Elevation  is  a  vertical  angle,  one  side  of 
which  is  horizontal,  and  the  other  inclined  upward  from  the 
angular  point. 

89.  An  Angle  of  Depression  is  a  vertical  angle,  one  side  of 
which  is  horizontal,  and  the  other  inclined  downward  from  the 
angular  point. 

In  Compass  and  Transit  Surveying,  in  addition  to  the  meas- 
urement of  lines,  angles  are  observed ;  hence,  besides  the 
instruments  previously  described,  we  present  the  following : 


THE  SURVEYOR'S  COMPASS. 

90.  The  Surveyor's  *  Compass  consists  essentially  of  a  brass 
plate  carrying  a  horizontal  graduated  circle,  in  the  centre  of 
which  is  suspended,  so  as  to  turn  freely,  a  magnetic  needle ; 
and  nt  the  extremities  of  the  plate  are  attached  vertically  two 
flattened  pieces  of  brass,  called  sights,  having  fine  slits  and 

*  The  Solar  Compass  is  described  in  Chapter  VI. 


56  PLANE   SURVEYING. 

circular  openings  in  them,  by  which  the  instrument  is  directed 
upon  any  object  or  station. 

In  addition  to  the  essentials  named,  this  instrument  usually 
has  two  small  spirit  levels  set  on  the  plate  at  right  angles  to 
each  other,  a  vernier  scale  for  setting  off  the  declination  of  the 
needle,  a  tangent  scale  for  reading  vertical  angles,  and  a  brass 
head  for  mounting  the  instrument  upon  a  tripod  or  a  single  staff 
called  Jacob's  Staff. 

91  The  graduated  circle  is  divided  into  half-degrees,  and  is 
figured  from  0  to  90  on  each  side  of  the  centre  line  of  zeros. 

The  magnetic  needle  is  from  4  to  6  inches  long  in  the  different 
sizes  of  compasses,  having  set  in  its  centre  a  piece  of  hardened 
steel  highly  polished,  which,  resting  upon  the  hardened  point  of 
the,  centre-pin,  allows  the  needle  to  turn  freely,  horizontally, 
and  to  take  its  direction  in  the  magnetic  meridian. 

92.  The  needle  is  lifted  from   its  support  by  a  concealed 
spring  actuated  by  a  screw.     The  test  of  the  delicacy   of  a 
magnetic  needle  is  the  number  of  vibrations  which  it  will  make 
in  a  certain  arc  before  coming  to  rest. 

When  the  compass  is  not  in  use,  the  needle  should  be  screwed 
up  against  the  glass,  and  the  instrument  set  so  that  the  north 
end  of  the  needle  points  towards  the  north. 

To  ADJUST  THE  COMPASS. 

93.  The  Levels.   First  bring  the  bubbles  into  the  centre,  by 
the  pressure  of  the  hand  on  different  parts  of  the  plate,  and 
then  turn  the  compass  half-way  around  ;    should  the  bubbles 
run  to  the  end  of  the  tubes,  it  would  indicate  that  those  ends 
were  the  highest :  lower  them  by  tightening  the  screws  immedi- 
ately under,  and  loosening  those  under  the  lowest  ends  until, 
by  estimation,  the  error  is  half  removed  ;  level  the  plate  again, 
and  repeat  the  first  operation  until  the  bubbles  will  remain  in 
the  centre  during  an  entire  revolution  of  the  compass. 


SURVEYOR'S    COMPASS. 


DESCRIPTION   OF   INSTRUMENTS.  59 

94.  The  Sights  may  next  be  tested  by  observing  through  the 
slits  a  fine  hair  or  thread,  made  exactly  vertical  by  a  plumb. 
Should  the  hair  appear  on  one  side  of  the  slit,  the  sight  must 
be  adjusted  by  filing  off  its  under  surface  on  that  side  which 
seems  the  highest. 

95.  The  Needle  is  adjusted  in  the  following  manner :  Having 
;he  eye  nearly  in  the  same  plane  with  the  graduated  rim  of  the 
compass-circle,  with  a  small  splinter  of  wood  or  a  slender  iron 
wire  bring  one  end  of  the  needle  in  line  with  any  prominent 
division  of  the  circle,  as  the  zero  or  ninety-degree  mark,  and 
notice  if  the  other,  end  corresponds  with  the  degree  on  the 
opposite  side:  if  it  does,  the  needle  is  said  to  "cut"  opposite 
degrees ;  if  not,  bend  the  centre-pin  by  applying  a  small  brass 
wrench,  about  one-eighth  of  an  inch  below  the  point  of  the  pin, 
until  the  ends  of  the  needle  are  brought  into  line  with  the  oppo- 
site degrees. 

Then,  holding  the  needle  in  the  same  position,  turn  the  com- 
pass half-way  around,  and  note  whether  it  now  cuts  opposite 
degrees ;  if  not,  correct  half  the  error  by  bending  the  needle, 
and  the  remainder  by  bending  the  centre-pin. 

The  operation  should  be  repeated  until  perfect  reversion  is 
secured  in  the  first  position. 

This  being  obtained,  it  may  be  tried  on  another  quarter  of 
the  circle  ;  if  any  error  is  there  manifested,  the  correction  must 
be  made  in  the  centre-pin  only,  the  needle  being  already 
straightened  by  the  previous  operation. 

96.  Electricity.  A  little  caution  is  necessary  in  handling  the 
compass,  that  the  glass  covering  be  not  excited  by  the  friction 
of  cloth,  silk,  or  the  hand,  so  as  to  attract  the  needle  to  its 
under  surface. 

When,  however,  the  glass  becomes  electrified,  the  charge 
may  be  removed  by  breathing  upon  it,  or  touching  different 
parts  of  its  surface  with  the  moistened  finger. 


60  PLANE   SURVEYING. 

97.  The  Needle  is  remagnetized  as  follows : 

The  operator,  being  provided  with  an  ordinary  permanent 
magnet,  and  holding  it  before  him,  should  pass  with  a  gentle 
pressure  each  end  of  the  needle  from  centre  to  extremity  over 
the  magnetic  pole,  describing  before  each  pass  a  circle  of  about 
six  inches  radius,  to  which  the  surface  of  the  pole  is  tangent, 
drawing  the  needle  towards  him,  and  taking  care  that  the 
north  and  the  south  ends  are  applied  to  the  opposite  poles  of 
the  magnet. 

Should  the  needle  be  returned  in  a  path  near  the  magnetic 
pole,  the  current  induced  by  the  contact  of  the  needle  and 
magnet,  in  the  pass  just  described,  would  be  reversed,  and 
thus  the  magnetic  virtue  almost  entirely  neutralized  at  each 
operation. 

When  the  needle  has  been  passed  about  twenty-five  times 
in  succession,  in  the  manner  just  described,  it  may  be  consid- 
ered as  fully  charged. 

A  fine  brass  wire  is  wound  in  two  or  three  coils  on  the  south 
end  of  the  needle,  and  may  be  moved  back  or  forth  in  order  to 
counterpoise  the  varying  weight  of  the  north  end. 

98.  The  Centre-Pin.   This  should  occasionally  be  examined, 
and  if  much  dulled,  taken  out  with  a  brass  wrench  or  with  a 
pair  of  pliers,  and  sharpened  on  a  hard  oil-stone  —  the  operator 
placing  it  in  the  end  of  a  small  stem  of  wood  or  a  pin-vise, 
and  delicately  twirling  it  with  the  fingers  as  he  moves  it  back 
and  forth  at  an  angle  of  about  30  degrees  to  the  surface  of  the 
stone. 

When  the  point  is  thus  made  so  fine  and  sharp  as  to  be  in- 
visible to  the  eye,  it  should  be  smoothed  by  rubbing  it  on  the 
surface  of  a  soft  and  clean  piece  of  leather. 

99.  Weight.  The  average  weights  of  the  different  sizes  of 
compasses,    including   the  brass   head  of  the  Jacob-staff,    be- 
ginning wth  the   smallest,    are   respectively  5£,   7£,  and   9^ 
pounds. 


DESCRIPTION   OF   INSTRUMENTS.  61 


THE  VERNIER. 

100.  A  Vernier  is  an  auxiliary  scale  for  measuring  smaller 
divisions  than  those  into  which  a  graduated  scale  or  limb  is 
divided.*  The  smallest  reading  of  the  vernier,  or  least  count, 
is  the  difference  in  length  between  one  division  on  the  gradu- 
ated scale  or  limb,  and  one  on  the  vernier.  If  the  divisions  on 
the  vernier  are  smnller  than  those  on  the  limb,  the  vernier  is 
direct;  if  the  reverse,  retrograde. 

6  7          8          9      •  10         11         12        13         14         15         16         17 


Let  LM  represent  any  scale  divided  into  tenths,  and  we  wish 
to  measure  or  read  to  tenths  of  these  divisions,  i.e.  to  y^v. 
Using  a  direct  vernier,  we  should  have  10  spaces  on  it  equal  to 
9  on  the  scale,  and  each  one  of  them  equal  to  T9¥  of  y1^,  or  y^, 
of  the  scale  graduation  ;  giving  a  least  count  of  y1^  —  Tf  7  =  y-^, 
as  desired.  To  read  to  twentieths  of  the  divisions  on  the  scale, 
we  should  have  20  divisions  on  the  vernier  corresponding  to  19 
on  the  scale,  or  each  space  on  the  vernier  equal  to  if '  iV  = '3inr> 
and  giving  a  least  count  of  -ffo  —  ^fa  =  -^fa. 

In  general,  if  s  =  the  smallest  division  of  the  scale  or  limb, 
v  =  the  smallest  division  of  the  vernier, 
n  =  number  of  divisions  on  the  vernier, 

we  shall  have  least  count  =  s  —  v  =  -• 

n 

Or,  the  least  count  of  a  vernier  is  equal  to  the  smallest  division 
of  the  scale  or  limb  divided  by  the  number  of  divisions  on  the 
vernier.f 

If  s  =  %  degree,  and  n  =  30,  as  ordinarily  found  on  transit 

*  It  derives  its  name  from  Peter  Vernier,  1631. 

t  It  is  evidently  immaterial  whether  LM  be  straight  or  curved. 


62  PLANE   SURVEYING. 

plates,  the  least  count  will  be  £-$-30  =  ^  of  a  degree  =  one 
minute. 

If  s  =  $  degree,  and  n  =  40,  oftentimes  found  on  vertical 
arcs  to  solar  attachments,  the  smallest  reading  =  \  -H  40  =  T|7 
of  a  degree  =  £  minute. 

To  space  a  vernier  for  a  given  least  count,  say  10",  on  a  limb 

graduated   to   10',  we  must  have  n  =  - =  —  =  60  spaces, 

s-v       | 
covering  59  spaces  on  the  limb. 

101.  To  read  an  Instrument  having  a  vernier  consists  in 
determining  the  number  of  units  and  fractional  parts  thereof, 
into  which  its  scale  or  limb  may  be  divided,  from  the  zero  point 
on  the  limb,  where  the  graduation  begins,  to  the  zero  point  of 
the  vernier. 

It  is  accomplished  as  follows :  Take  the  reading  of  the  scale, 
as  shown  by  the  last  graduation  preceding  the  zero  of  the  ver- 
nier ;  then  find  a  line  on  the  vernier  which  coincides  with  a  line 
on  the  scale.  The  number  of  this  line,  as  indicated  by  the 
graduation  on  the  vernier,  shows  how  many  units  of  the  least 
count  are  to  be  added  to  the  first  reading. 

EXERCISES. 

1.  A  levelling-rod  is  graduated  into  feet,  tenths,  and  hun- 
dredths.     It  is  required  to  space  a  direct  vernier  so  that  the 
rod  may  be  read  to  thousandths  of  a  foot. 

2.  An  arc  is  graduated  into  quarter-degrees,  and  a  vernier 
of  30  parts  covers  29  parts  of  the  arcs  ;  find  the  least  count. 

3.  A  scale  is  divided  into  inches  and  tenths  of  an  inch  ;  plan 
a  direct  vernier  by  means  of  which  the  scale  may  be  read  to 
j-l^  of  an  inch. 

Plan  a  retrograde  vernier  to  accomplish  the  same  object. 

4.  Design  a  vernier  which  when  applied  to  a  limb  graduated 
into  20'  will  give  a  least  count  of  20". 


SURVEYOR'S   TRANSIT. 

NOTE.  The  principal  part  of  the  description  of  the  Compass  and  Tran- 
sit, and  the  plates  for  the  engraving  of  these  instruments,  were  kindly 
furnished  by  Messrs.  W.  &  L.  E.  Gurley,  Troy,  X.Y. 


DESCRIPTION   OF  INSTRUMENTS. 


65 


SURVEYOR'S  TRANSIT. 

102.  The  essential  parts  of  the  Tran- 
sit, as  shown  in  the  cut,  are  the  telescope 
with  its  axis  and  two  supports,  the  cir- 
cular plates  with  their  attachments,  the 
sockets  upon  which  the  plates  revolve, 
the  levelling-head,  and  the  tripod  on 
which  the  whole  instrument  stands. 

The  telescope  is  from  10  to  11  inches 
long,  firmly  secured  to  an  axis  having 
its  bearings  nicely  fitted  in  the  stand- 
ards, and  thus  enabling  the  telescope 
to  be  moved  in  either  direction,  or 
turned  completely  around  if  desired. 

The  different  parts  of  the  telescope 
are  shown  in  the  marginal  figure. 

The  object-glass  is  composed  of  two 
lenses,  so  as  to  show  objects  without 
color  or  distortion,  is  placed  at  the  end 
of  a  slide  having  two  bearings,  one  at 
the  end  of  the  outer  tube,  the  other  in 
the  ring  (7(7,  suspended  within  the  tube 
by  four  screws,  only  two  of  which  are 
shown  in  the  cut. 

The  object-glass  is  carried  out  or  in 
by  a  pinion  working  in  a  rack  attached 
to  the  slide,  and  thus  adjusted  to  ob- 
jects either  near  or  remote  as  desired. 

The  eye-piece  is  made  up  of  four 
plano-convex  lenses,  which,  beginning 
at  the  eye-end,  are  called  respectively 
the  eye,  the  field,  the  amplifying,  and 
the  object-lenses,  the  whole  forming  a 
compound  microscope  having  its  focus 
in  the  plane  of  the  cross-wire  ring  BB. 


66  PLANE   SURVEYING. 

The  eye-piece  is  brought  to  its  proper  focus  usually  by  twist- 
ing its  milled  end,  the  spiral  movement  within  carrying  the  eye- 
tube  out  or  in  as  desired  ;  sometimes  a  pinion,  like  that  which 
focuses  the  object-glass,  is  employed  for  the  same  purpose. 

103.  The  Cross-Wires  are  two  fibres  of  spider-web  or  very 
fine  platinum  wire,  cemented  into  the  cuts  on  the  surface  of  a 


metal  ring,  at  right  angles  to  each  other,  so  as  to  divide  the 
open  space  in  the  centre  into  quadrants. 

104.  Optical  Axis.    The  intersection  of  the  wires  forms  a 
very  minute  point,  which,  when  the\r  are  adjusted,  determines 
the  optical  axis  of  the  telescope,  and  enables  the  surveyor  to 
fix  it  upon  an  object  with  the  greatest  precision. 

The  imaginary  line  passing  through  the  optical  axis  of  the 
telescope  is  termed  the  "line  of  collimation,"  and  the  opera- 
tion of  bringing  the  intersection  of  the  wires  into  the  optical 
axis,  is  called  the  "adjustment  of  the  line  of  collimation." 
This  will  be  hereafter  described. 

105.  The  Standards  of  the  Transit  are  firmly  attached  by 
their  expanded  bases  to  the  upper  plate,  one  of  them  having 
near  the  top,  as  shown  in  the  cut,  a  little  movable  box,  actu- 
ated by  a  screw  underneath,  by  which  the  telescope  axis  is  made 
truly  horizontal,  as  will  be  hereafter  described. 


DESCRIPTION   OF   INSTRUMENTS. 


67 


The  sectional  view  here  given  shows  the  interior  construction 
of  the  sockets  of  the  transit,  the  manner  in  which  it  is  detached 
from  the  spindle,  and  the  means  by  which  it  can  be  taken  apart 
if  desired. 

In  the  figure,  the  limb  BB  is  attached  to  the  main  socket  C, 
which  is  itself  carefully  fitted  to  the  conical  spindle  //,  and  held 
in  place  by  the  spring  catch  S. 


The  upper  plate,  AA,  carrying  the  compass-circle,  standards, 
etc.,  is  fastened  to  the  flanges  of  the  socket  7f,  which  is  fitted 
to  the  upper  conical  surface  of  the  main  socket  (7;  the  weight 
of  all  the  parts  being  supported  on  the  small  bearings  of  the 
end  of  the  socket,  as  shown,  so  as  to  turn  with  the  least  possible 
friction. 

A  small  conical  centre,  in  which  from  below  is  inserted  a 
strong  screw,  is  brought  down  firmly  upon  the  upper  end  of 
the  main  socket  (7,  and  thus  holds  the  two  plates  of  the 
instrument  securely  together,  while  at  the  same  time  allowing 
them  to  move  freely  around  each  other  in  use. 


68  PLANE    SURVEYING. 

A  small  disc  above  the  conical  centre  contains  the  steel  cen- 
tre-pin upon  which  rests  the  needle,  as  shown  ;  the  disc  is  fas- 
tened to  the  upper  plate  by  two  small  screws,  as  represented. 

The  main  socket  with  all  its  parts  is  of  the  best  bell-metal 
and  is  most  carefully  and  thoroughly  made,  the  long  bearing  of 
the  sockets  insuring  their  firm  and  easy  movement,  while  at 
the  same  time  they  are  entirely  out  of  the  reach  of  dust,  or 
other  source  -of  wear. 

When  desii'ed,  the  whole  upper  part  of  the  instrument  can  be 
taken  off  from  the  spindle  by  pulling  out  the  head  of  the  spring 
catch  at  S,  and  when  replaced  will  be  secured  by  the  self-acting 
spring  of  the  catch. 

The  figure  also  shows  the  covers  of  the  levelling-screws,  the 
shifting  centre  of  the  lower  le veiling-plate,  and  the  screw  and 
loop  for  the  attachment  of  the  plummet. 

The  compass-box,  containing  the  needle,  etc.,  is  covered  by 
a  glass  to  exclude  the  moisture  and  air;  the  circle  is  silvered, 
and  is  divided  on  its  upper  surface  or  rim  into  degrees  and 
half-degrees,  the  degree  marks  being  also  cut  down  on  its  inner 
edge,  and  figured  from  0  to  90  on  each  side  of  the  centre  or 
line  of  zero. 

106.  The  Magnetic  Needle  is  four  to  five  inches  long  in  the 
different  sizes  of  transits,  its  brass  cap  having  inserted  in  it  a 
little  socket  or  centre  of  hardened  steel,  perfectly  polished,  and 
this  resting  upon  the  hardened  and  polished  point  of  the  centre- 
pin,  allows  the  needle  to  play  freely  in  a  horizontal  direction, 
and  thus  take  its  direction  in  the  magnetic  meridian.  The 
needle  has  its  north  end  designated  by  a  scallop  or  other  mark, 
and  on  its  south  end  'a  small  coil  of  fine  brass  wire,  easily 
moved,  so  as  to  bring  both  ends  of  the  needle  to  the  same 
level.  The  needle  is  lifted  from  the  pin  by  a  concealed  spring 
underneath  the  upper  plate,  actuated  by  a  screw  shown  above, 
thus  raising  the  button  so  as  to  check  the  vibrations  of  the 
needle,  or  bring  it  up  against  the  glass  when  not  in  use,  to 
avoid  the  unnecessary  wear  of  the  pivot. 


DESCRIPTION   OF   INSTRUMENTS.  69 

107.  The  Clamp  and  Tangent  Movement,  shown  in  the  en- 
graving, page  64,  attached  to  the  plates,  serves  to  fasten  the  two 
plates  together,  so  that  by  the  tangent  screw  they  can  be  slowly 
moved  around  each  other  in  either  direction,  or  loosened  at  will 
and  moved  by  the  hand,  thus  enabling  one  to  direct  the  tele- 
scope rapidly  and  accurately  to  the  point  of  sight. 

The  Two  Levels  are  shown  placed  at  right  angles  to  each  other 
so  as  to  level  the  plate  in  all  directions,  and  adjusted  by  turn- 
ing the  capstan-head  screws  at  their  ends,  by  a  small  steel 
adjusting-pin.  The  glass  vials  used  in  the-  levels  are  ground 
on  their  upper  interior  surface,  so  as  to  make  the  bubble  move 
evenly  and  with  great  sensitiveness. 

108.  The  Lower  Plate,  or  Limb  BB,  is  divided  on  its  upper 
surface — usually  into  degrees  and  half-degrees  —  and  generally 
figured  in  two  rows  ;  viz.,  from  0  to  360,  and  from  0  to  90  each 
way. 

109.  The  Verniers  are  double,  having  on  each  side  of  the 
zero  mark  thirty  equal  divisions  corresponding  precisely  with 
twenty-nine  half-degrees  of  the  limb  ;  they  thus  read  to  single 
minutes,  and  the  number  passed  over  is  counted  in  the  same 
direction  in  which  the  vernier  is  moved. 

The  use  of  two  opposite  verniers  in  this  and  other  instru- 
ments gives  the  means  of  "  cross-questioning"  the  graduations, 
the  perfection  with  which  they  are  centred,  and  the  dependence 
which  can  be  placed  upon  the  accuracy  of  the  angles  indicated. 

Reflectors  of  silver  or  celluloid,  as  in  the  mountain  transit, 
are  often  used  to  throw  more  light  upon  the  divisions,  and  more 
rarely  shades  of  ground  glass  are  employed  to  give  a  clear  but 
more  subdued  light. 

110.  The  Graduations  are  made  commonly  on  the  brass  sur- 
face of   the  limb,  afterwards  filled  with  black  wax,  and  then 
finished    and   silvered.      Many    instruments,   however,  have    a 
solid  silver  plate  put  over  the  brass,  and  the  graduations  made 
on  the  silver  itself. 


70  PLANE   SURVEYING. 

The  last  is  more  costly,  but  insures  a  finer  graduation,  with 
less  liability  to  tarnish  or  change  color. 

111.  The  Sockets  of  the  transit  are  compound ;  the  interior 
spindle  attached  to  the  vernier  plate,  turning  in  the  exterior 
socket  C  when  an  angle  is  taken  on  the  limb ;  but  when  the 
plates  are  clamped  together,  the  exterior  socket  itself,  and  with 
it  the  whole  instrument,  revolves  in  the  socket  of  the  levelling- 
head. 

The  sockets  are  made  with  the  greatest  care,  the  surfaces 
being  truly  concentric  with  each  other,  and  the  bell-metal  or 
composition  of  which  the}"  are  composed,  of  different  degrees 
of  hardness,  so  as  to  cause  them  to  move  upon  each  other  easily 
and  with  the  least  possible  wear. 

The  levelling-head  also  consists  of  two  plates  connected  to- 
gether by  a  socket,  having  at  its  end  a  hemispherical  nut,  fitting 
into  a  corresponding  cavity  in  the  lower  plate. 

The  plates  are 'inclined  to  each  other  or  made  parallel  at  will 
by  four  levelling-screws,  of  which  only  two  are  shown  in  the 
section. 

The  screws  are  of  bronze  or  hard  composition  metal  and  fitted 
to  long  nuts  of  brass,  screwed  into  the  upper  parallel  plate ; 
and,  as  will  be  noticed,  have  threads  only  on  the  upper  ends, 
the  lower  part  of  their  steins  turning  closely  in  the  lower  un- 
threaded part  of  the  nuts. 

By  this  arrangement  dust  is  excluded  from  the  lower  end  of 
the  screws,  while  the  brass  cover  above  equally  protects  the 
other  end. 

The  screws  rest  in  little  cups  or  sockets,  which  are  secured 
to  their  ends  and  in  which  they  turn  without  marring  the  sur- 
face of  the  lower  plate,  the  cups  also  permitting  the  screws  to 
be  shifted  from  side  to  side,  or  turned  around  in  either  direc- 
tion on  the  lower  plate. 

The  clamp  and  tangent  movement  of  the  levelling-head  serves 
to  turn  the  whole  instrument  upon  its  sockets,  so  as  to  fix  the 
telescope  witli  precision  upon  any  given  point,  and  when  un- 


DESCRIPTION    OF   INSTRUMENTS.  71 

clamped  allowing  it  to  be  directed  approximately  by  hand. 
The  tangent  screws,  as  will  be  seen,  press  on  opposite  sides  of 
the  clamp-piece,  and  thus  insure  a  very  fine  and  solid  movement 
of  the  instrument. 

112.  The  Lower  Levelling-Plate  is  made  in  two  pieces  —  the 
upper  one,  which  is  screwed  fast  to  the  top  of  the  tripod,  having 
a  large  opening  in  its  centre,  in  which  the  smaller  lower  one  is 
shifted  from  side  to  side,  or  turned  completely  around. 

By  this  simple  arrangement,  termed  a  shifting  centre,  the 
instrument  is  easily  moved  over  the  upper  plate,  and  the  plum- 
met which  hangs  from  the  centre  P,  set  precisely  over  a  point, 
without  moving  the  tripod. 

113.  The  Levelling-Head  of  the  engineer's  transit  is  attached 
to  the  sockets  by  a  screw  and  washer  below ;  it  can  be  removed 
for  cleaning,  oiling,  etc.,  but  should  be  in  place  when  the  in- 
strument is  in  use,  or  packed  for  transportation. 

114.  The  Tripod  has  three  mahogany  legs,  the  upper  ends 
of  which  are  pressed  firmly  on  each  side  of  a  strong  tenon  on 
the  solid  bronze  head  by  a  bolt  and  nut  on  opposite  sides  of  the 
leg ;  the  nut  can  also  be  screwed  up  at  will  by  a  wrench  fur- 
nished for  the  purpose,  and  thus  kept  firm. 

The  lower  end  of  the  leg  has  a  brass  shoe  with  iron  point, 
securely  fastened  and  riveted  to  the  wood. 

115.  To  Adjust  the  Transit.      Every  instrument  should  leave 
the  hands  of  the  maker  in  complete  adjustment ;  but  all  are  so 
liable  to  derangement  by  accident  or  careless  use,  that  we  deem 
it  necessary  to  describe  particularly  those  which  are  most  likely 
to  need  attention. 

The  principal  adjustments  of  the  transit  are  : 

1 .  The  Levels. 

2.  The  Line  of  Collimation. 

3.  The  Standards. 


72  PLANE   SURVEYING. 

116.  To  Adjust  the  Levels.     Set  up  the  instrument  upon  its 
tripod  as  nearly  level  as  may  be,  and  having  undamped  the 
plates,  bring  the  two  levels  above  and  on  a  line  with  the  two 
pairs  of  levelling-screws ;  then,  with  the  thumb  and  first  finger 
of  each  hand  clasp  the  heads  of  two,  opposite ;    and,  turning 
both  thumbs  in  or  out,  as  may  be  needed,  bring  the  bubble  of 
the  level  directly  over  the  screws,  exactly  to  the  centre  of  the 
opening.     Without  moving  the  instrument,  proceed  in  the  same 
manner  to  bring  the  other  bubble  to  its  centre  ;    after  doing 
this,  the  level  first  corrected  may  be  thrown  a  little  out ;  bring 
it  in  again  ;  and  when  both  are  in  place,  turn  the  instrument 
half-way  around :  if  the  bubbles  both  come  to  the  centre,  they 

•would  need  no  correction,  but  if  not,  with  the  adjusting-pin 
turn  the  small  screws  at  the  end  of  the  levels  until  the  bubbles 
are  moved  over  half  the  error ;  then  bring  the  bubbles  again 
into  the  centre  by  the  levelling-screws,  and  repeat  the  operation 
until  the  bubbles  will  remain  in  the  centre  during  a  complete 
revolution  of  the  instrument,  and  the  adjustment  will  be 
complete. 

117.  To  Adjust  the  Line  of  Collimation.     To  make  this 
adjustment,  —  which    is,   in   other  words,   to  bring  the  inter- 
section of  the  wires  into  the  optical  axis  of  the  telescope,  so 
that  the  instrument,  when  placed  in  the  middle  of  a  straight 
line,  will,  by  the  revolution  of  the  telescope,  cut  its  extremities, 
—  proceed  as  follows : 

Set  the  instrument  firmly  on  the  ground  and  level  it  care- 
fully ;  and  then,  having  brought  the  wires  into  the  focus  of  the 
eye-piece,  adjust  the  object-glass  on  some  well-defined  point, 
as  the  edge  of  a  chimney  or  other  object,  at  a  distance  of  from 
200  to  500  feet ;  determine  if  the  vertical  wire  is  plumb,  by 
clamping  the  instrument  firmly  and  applying  the  wire  to  the 
vertical  edge  of  a  building,  or  observing  if  it  will  move  parallel 
to  a  point  taken  a  little  to  one  side  :  should  any  deviation  be 
manifested,  loosen  the  cross-wire  screws,  and  by  the  pressure 
of  the  hand  on  the  head  outside  the  tube,  move  the  ring  around 
until  the  error  is  corrected. 


DESCRIPTION   OF   INSTRUMENTS.  73 

The  wires  being  thus  made  respectively  horizontal  and 
vertical,  fix  their  point  of  intersection  on  the  object  selected ; 
clamp  the  instrument  to  the  spindle,  and  having  revolved  the 
telescope,  find  or  place  some  good  object  in  the  opposite  direc- 
tion, and  at  about  the  same  distance  from  the  instrument  as 
the  first  object  assumed. 

Great  care  should  always  be  taken  in  turning  the  telescope, 
that  the  position  of  the  instrument  upon  the  spindle  is  not  in 
the  slightest  degree  disturbed. 

Now,  having  found  or  placed  an  object  which  the  vertical 
wire  bisects,  unclamp  the  instrument,  turn  it  half-way  around, 
and  direct  the  telescope  to  the  first  object  selected ;  having 
bisected  this  with  the  wires,  again  clamp  the  instrument, 
revolve  the  telescope,  and  note  if  the  vertical  wire  bisects  the 
second  object  observed. 

Should  this  happen,  it  will  indicate  that  the  wires  are  in 
adjustment,  and  the  points  bisected  are  with  that  of  the  centre 
of  the  instrument,  in  the  same  straight  line. 

If  not,  however,  the  space  which  separates  the  wires  from 
the  second  point  observed,  will  be  double  the  deviation  of  that 
point  from  a  true  straight  line,  which  may  be  conceived  as 
drawn  through  the  first  point  and  the  centre  of  the  instrument, 
since  the  error  is  the  result  of  two  observations,  made  with  the 
wires  when  they  are  out  of  the  optical  axis  of  the  telescope. 


For,  as  in  the  diagram,  let  A  represent  the  centre  of  the 
instrument,  and  BC  the  imaginary  straight  line,  upon  the  ex- 
tremities of  which  the  line  of  collimation  is  to  be  adjusted. 

B  represents  the  object  first  selected,  and  D  the  point  which 
the  wires  bisected,  when  the  telescope  was  made  to  revolve. 

When  the  instrument  is  turned  half  around,  and  the  telescope 
again  directed  to  .B,  and  once  more  revolved,  the  wires  will 


74  PLANE   SURVEYING. 

bisect  an  object  -E,  situated  as  far  to  one  side  of  the  true  line 
as  the  point  D  is  on  the  other  side. 

The  space  DE,  is  therefore  the  sum  of  two  deviations  of  the 
wires  from  a  true  straight  line,  and  the  error  is  made  very 
apparent. 

In  order  to  correct  it,  use  the  two  capstan-head  screws  on 
the  sides  of  the  telescope,  these  being  the  ones  which  affect  the 
position  of  the  vertical  wire. 

Remember  that  the  eye-piece  inverts  the  position  of  the 
wires,  and  therefore,  that  in  loosening  one  of  the  screws  and 
tightening  the  other  on  the  opposite  side,  the  operator  must 
proceed  as  if  to  increase  the  error  observed.  Having  in  this 
manner  moved  back  the  vertical  wire  until,  by  estimation,  one- 
quarter  of  the  space  DE  has  been  passed  over,  return  the 
instrument  to  the  point  jB,  revolve  the  telescope,  and  if  the 
correction  has  been  carefully  made,  the  wires  will  now  bisect  a 
point  C,  situated  midway  between  D  and  E,  and  in  the  pro- 
longation of  the  imaginary  line,  passing  through  the  point  B 
and  the  centre  of  the  instrument. 

To  ascertain  if  such  is  the  case,  turn  the  instrument  half 
around,  fix  the  telescope  upon  B,  clamp  to  the  spindle,  and 
again  revolve  the  telescope  towards  C.  If  the  wires  again 
bisect  it,  it  will  prove  that  they  are  in  adjustment,  and  that 
the  points  B,  A,  C,  all  lie  in  the  same  straight  line. 

Should  the  vertical  wire  strike  to  one  side  of  (7,  the  error 
must  be  corrected  precisely  as  above  described,  until  it  is 
entirely  removed. 

118.  To  Adjust  the  Standards.  In  order  that  the  wires  may 
trace  a  vertical  line  as  the  telescope  is  moved  up  or  down,  it  is 
necessary  that  both  the  standards  of  the  telescope  should  be  of 
precisely  the  same  height. 

To  ascertain  this  and  make  the  correction  if  needed,  proceed 
as  follows : 

Having  the  line  of  collimation  previously  adjusted,  set  up  the 
instrument  in  a  position  where  points  of  observation,  such  as 


DESCRIPTION   OF    INSTRUMENTS.  75 

the  point  and  base  of  a  lofty  spire,  can  be  selected,  giving  a 
long  range  in  a  vertical  direction. 

Level  the  instrument,  fix  the  wires  on  the  top  of  the  object, 
and  clamp  to  the  spindle ;  then  bring  the  telescope  down,  until 
the  wires  bisect  some  good  point,  either  found  or  marked  at  the 
base ;  turn  the  instrument  half  around,  fix  the  wires  on  the 
lower  point,  clamp  to  the  spindle,  and  raise  the  telescope  to 
the  highest  object. 

If  the  wires  bisect  it,  the  vertical  adjustment  is  effected ;  if 
they  are  thrown  to  either  side,  this  would  prove  that  the  stand- 
ard opposite  that  side  was  the  highest,  the  apparent  error  being 
double  that  actually  due  to  this  cause. 

To  correct  it,  one  of  the  bearings  of  the  axis  is  made  mov- 
able, so  that  by  turning  a  screw  underneath  this  sliding  piece, 
as  well  as  the  screws  which  hold  on  the  cap  of  the  standard,  the 
adjustment  is  made  with  the  utmost  precision. 

OTHER  ADJUSTMENTS  OF  THE  TRANSIT. 

Besides  the  three  adjustments  already  described  —  which  are 
all  that  the  surveyor  will  ordinarily  have  to  make  —  there  are 
those  of  the  needle  and  the  object-glass  slide  which  may  some- 
times be  required. 

The  first  is  given  with  the  description  of  the  compass ;  the 
last  will  now  be  described. 

119.  To  Adjust  the  Object-Slide.  Having  set  up  and  levelled 
the  instrument,  the  line  of  collimation  being  also  adjusted  for 
objects  from  300  to  500  feet  distant,  clamp  the  plates  securely, 
and  fix  the  vertical  cross-wire  upon  an  object  as  distant  as  may 
be  distinctly  seen  ;  then,  without  disturbing  the  instrument, 
throw  out  the  object-glass,  so  as  to  bring  the  vertical  wire  upon 
an  object  as  near  as  the  range  of  the  telescope  will  allow. 
Having  this  clearly  in  mind,  unclamp  the  limb,  turn  the  instru- 
ment half-way  around,  reverse  the  eye-end  of  the  telescope, 
clamp  the  limb,  and  with  the  tangent-screw  bring  the  vertical 


76  PLANE   SURVEYING. 

wire  again  upon  the  near  object ;  then  draw  in  the  object-glass 
slide  until  the  distant  object  first  sighted  upon  is  brought  into 
distinct  vision.  If  the  vertical  wire  strikes  the  same  line  as  at 
first,  the  slide  is  correct  for  both  near  and  remote  objects  ;  and, 
being  itself  straight,  for  all  distances. 

But  if  there  be  an  error,  proceed  as  follows :  first,  with  the 
thumb  and  forefinger  twist  off  the  thin  brass  tube  that  covers 
the  screws  CC  shown  in  the  sectional  view  of  the  telescope, 
p.  65.  Next,  with  the  screw-driver,  turn  the  two  screws  CO 
on  the  opposite  ides  of  the  telescope,  loosening  one  and  tight- 
ening the  other,  so  as  apparently  to  increase  the  error,  making, 
by  estimation,  one-half  the  correction  required. 

Then  go  over  the  usual  adjustment  of  the  line  of  collimation , 
and  having  it  completed,  repeat  the  operation  above  described ; 
first  sighting  upon  the  distant  object,  then  finding  a  near  one 
in  line,  and  then  reversing,  making  correction,  etc.,  until  the 
adjustment  is  complete. 

120.  To  Use  the  Transit.  The  instrument  should  be  set  up 
firmly,  the  tripod  legs  being  pressed  into  the  ground,  so  as  to 
bring  the  plates  as  nearly  level  as  convenient ;  the  plates  should 
then  be  carefully  levelled  and  property  clamped,  the  zeros  of  the 
verniers  and  limb  brought  into  line  by  the  upper  tangent-screw, 
and  the  telescope  directed  to  the  object  by  the  tangent-screws 
of  levelling-head. 

The  angles  taken  are  then  read  off  upon  the  limb,  without 
subtracting  from  those  given  by  the  verniers,  in  any  other 
position. 

Before  an  observation  is  made  with  the  telescope,  the  eye- 
piece should  be  moved  in  or  out,  until  the  wires  appear  distinct 
to  the  eye  of  the  operator ;  the  object-glass  is  then  adjusted  by 
turning  the  pinion-head  until  the  object  is  seen  clear  and  well- 
defined,  and  the  wires  appear  as  if  fastened  to  its  surface. 

The  intersection  of  the  wires,  being  the  means  by  which  the 
optical  axis  of  the  telescope  is  defined,  should  be  brought  pre- 
cisely upon  the  centre  of  the  object  to  which  the  instrument  is 
directed. 


DESCRIPTION    OF   INSTRUMENTS.  77 

The  needle  is  used,  as  in  the  compass,  to  give  the  bearing  of 
lines,  and  as  a  rough  check  upon  the  angles  obtained  by  the 
verniers  and  limb ;  but  its  employment  is  only  subsidiary  to  the 
general  purposes  of  the  transit. 

121.  Attachments  of  Transits.   The  engraving  of  the  Sur- 
veyor's Transit  represents  the  attachments  often  applied  to  the 
Engineer's  Transit,   viz. :    vertical   circle,  level  on  telescope, 
and  clamp  and  tangent  to  telescope  axis.      They  are  of  use 
where  approximate  levelling  and  vertical  angles  are  to  be  taken 
in  connection  with  the  ordinary  use  of  the  transit,  and  with 
their  adjustments,  etc.,  will  now  be  described. 

122.  The  Vertical  Circle  firmly  secured  to  the  axis  of  the 
telescope  is  4^  inches  in  diameter,  plated  with  silver,  divided  to 
hnlf-degrees,  and  with  its  vernier  enables  the  surveyor  to  obtain 
vertical  angles  to  single  minutes. 

123.  The  Level  on  Telescope  consists  of  a  brass  tube  about 
6|  inches  long,  each  end  of  which  is  held  between  two  capstan- 
nuts  connected  with  a  screw  or  stem  attached  to  the  under  side 
of  the  telescope  tube. 

124.  The  Clamp  and  Tangent  consists  of  an  arm  at  one  end 
encircling  the  telescope  axis,  and  at  the  other  connected  with 
the  tangent-screw  ;  the  clamp  is  fastened  at  will  to  the  axis  by 
a  clamp-screw,  inserted  at  one  side  of  the  ring,  and  then  by 
turning  the  taugent-screw  the  telescope  is  raised  or  lowered  as 
desired. 

125.  To  Adjust  the  Vertical  Circle.    Having  the  instrument 
firmly  set  up  and  carefully  leveled,  bring  into  line  the  zeros  of 
the  circle  and  vernier,  and  with  the  telescope  find  or  place  some 
well-defined  point  or  line,  from  200  to  300  feet  distant,  which 
is  cut  by  the  horizontal  wire. 

Turn  the  instrument  half-way  around,  revolve  the  telescope, 
and  fixing  the  wire  upon  the  same  point  as  before,  note  if  the 
zeros  are  again  in  line. 


78  PLANE   SURVEYING. 

If  not,  loosen  the  capstan-head  screws,  which  fasten  the 
vernier,  and  move  the  zero  of  the  vernier  over  half  the  error ;  * 
bring  the  zeros  again  into  coincidence,  and  proceed  precisely  as 
at  first,  until  the  error  is  entirely  corrected,  when  the  adjust- 
ment will  be  complete. 

This  method  is  not  applicable  when  only  an  arc  of  a  circle  is 
attached.  The  adjustment  may  then  be  made  as  follows: 
Observe  successively  from  each  of  the  two  points  to  the  other, 
and  as  before  use  half  the  error  in  adjusting  the  vernier. 
Verifv  bty  repetition. 

A  slight  error  may  be  most  readily  removed  by  putting  the 
zeros  in  line  and  then  moving  the  wire  itself  over  half  the 
interval. 

126.  The  Level  is  Adjusted  by  bringing  the  bubble  carefully 
into  the  centre  by  the  nuts  at  each  end ;  and  when  there  is  a 
vertical  circle  on  the  instrument,  this  should  be  done  when  the 
zeros  of  circle  and  vernier  are  in  line  and  in  adjustment ;  when 
there  is  no  vertical  circle,  proceed  as  follows  : 

127.  To  Adjust  the  Level  on  Telescope.    Choose  a  piece  of 
ground  nearly  level,  and  having  set  the  instrument  firmly,  level 
the  plates  carefully,  and  bring  the  bubble  of  the  telescope  into 
the  centre  with  the  tangent-screw.     Measure  in  any  direction 
from  the  instrument,  from  100  to  300  feet,  and  drive  a  stake, 
and  on  the  stake  set  a  staff,  and  note  the  height  cut  by  the 
horizontal  wire ;  then  take  the  same  distance  from  the  instru- 
ment in  an  opposite  direction,  and  drive  another  slake. 

On  that  stake  set  the  staff,  and  note  the  height  cut  by  the 
wire  when  the  telescope  is  turned  in  that  direction. 

The  difference  of  the  two  observations  is  evidently  the  dif- 
ference of  level  of  the  two  stakes. 

Set  the  instrument  over  the  lowest  stake,  or  that  upon  which 


*  Called  Index  Error.     It  may  be  rectified  as  here  shown,  or  each  obser- 
vation corrected  by  this  amount. 


DESCRIPTION    OF    INSTRUMENTS.  79 

the  greatest  height  was  indicated,  and  bring  the  levels  on  the 
plates  and  telescope  into  adjustment  as  at  first. 

Then,  with  the  staff,  measure  the  perpendicular  distance  from 
the  top  of  the  stake  to  the  centre  of  one  of  the  horizontal  cross- 
wire  screw-heads  ;  from  that  distance  subtract  the  difference  of 
level  between  the  two  stakes  and  mark  the  point  on  the  staff 
thus  found  ;  place  the  staff  on  the  other  stake,  and  with  the 
tangent-screw  bring  the  horizontal  wire  to  the  mark  just  found, 
and  the  line  will  be  level. 

The  telescope  now  being  level,  bring  the  bubble  of  the  level 
into  the  centre,  by  turning  the  little  nuts  at  the  end  of  the  tube, 
and  noting  again  if  the  wires  cut  the  point  on  the  staff ;  screw 
up  the  nuts  firmly  and  the  adjustment  will  be  completed. 

128.  To  Take  Apart  the  Surveyor's   Transit.     When  it  is 
necessary  to  separate  the  plates  of  the   transit,    proceed   as 
follows : 

(1)  Remove  the  clamp-screw  and  take  off  the  head  of  the 
pinion,  both  on  the  north  end  and  outside  the  compass  circle. 

(2)  Unscrew  the  bezel  ring  containing  the  glass  cover  of  the 
compass,  remove  the  needle  and  button  beneath  it,  and  take 
out  the  two  small  screws  so  as  to  remove  the  disc. 

(3)  Take  the  instrument  from  its  spindle,  and  with  a  large 
screw-driver  take  out  the  screw  from  the  underside  of  the  coni- 
cal centre  (see  figure,  p.  67). 

(4)  Drive  out  the  centre  from  below  by  a  round  piece  of 
wood,  holding  the  instrument  vertical  so  that  the  centre  will 
not  bruise  the  circle. 

(5)  Set  the  instrument  again  upon  its  spindle,  take  out  the 
clamp-screw  to  the  tangent  movement  of  the  limb,  and  the  work 
is  complete.    To  put  the  transit  together  again,  proceed  exactly 
the  reverse  of  the  operation  thus  described. 

129.  The  Solar  Attachment  is  essentially  the  solar  apparatus 
of  Burt  placed  upon  the  cross-bar  of  the  ordinary  transit,  the 
polar  axis  only  being  directed  above  instead  of  below,  as  in  the 
solar  compass. 


80  PLANE   SURVEYING. 

A  little  circular  disc  of  an  inch  and  a  half  diameter,  and  hav- 
ing a  short,  round  pivot  projecting  above  its  upper  surface,  is 
first  securely  screwed  to  tho  telescope  axis. 

Upon  this  pivot  rests  the  enlarged  base  of  the  polar  axis,  which 
is  also  firmly  connected  with  the  disc  by  four  capstan-head 
screws  passing  from  the  under  side  of  the  disc  into  the  base 
already  named. 

These  screws  serve  to  adjust  the  polar  axis,  as  will  be  ex- 
plained hereafter. 

130.  The  Hour  Circle  surrounding  the  base  of  the  polar  axis 
is  easily  movable  about  it,  and  can  be  fastened  at  any  point 
desired  by  two  flat-head  screws  above.     It  is  divided  to  five 
minutes  of  time  ;  is  figured  from  I.  to  XII.,  and  is  read  by  a 
small  index  fixed  to  the  declination  circle,  and  moving  with  it. 

A  hollow  cone,  or  socket,  fitting  closely  to  the  polar  axis, 
and  made  to  move  snugly  upon  it,  or  clamped  at  any  point 
desired  by  a  milled-head  screw  on  top,  furnishes  by  its  two 
expanded  arms  below  a  firm  support  for  the  declination  arc, 
which  is  securely  fastened  to  it  by  two  large  screws,  as  shown. 

131.  The  Declination  Arc  is  of  about  5  inches  radius,  is 
divided  to  quarter  degrees,  and  reads  by  its  vernier  to  single 
minutes  of  arc,  the  divisions  of  both  vernier  and  limb  being  in 
the  same  plane. 

The  declination  arm  has  the  usual  lenses  and  silver  plates  on 
the  two  opposite  blocks,  made  precisely  like  those  of  the  ordi- 
nary solar  compass,  but  its  vernier  is  outside  the  block,  and 
more  easily  read. 

The  declination  arm  has  also  a  clamp  and  tangent  movement, 
as  shown  in  the  cut.  The  arc  of  the  declination  limb  is  turned 
on  its  axis,  and  one  of  the  other  solar  lens  used,  as  the  sun  is 
north  or  south  of  the  equator ;  the  cut  shows  its  position  when 
it  is  north. 

The  Latitude  is  set  off  by  means  of  a  large  vertical  limb  hav- 
ing a  radius  of  2|  inches  ;  the  arc  is  divided  to  twenty  minutes, 


TRANSIT  WITH   SOLAR  ATTACHMENT. 


DESCRIPTION   OF   INSTRUMENTS.  83 

is  figured  from  the  centre,  each  way,  up  to  80°,  and  is  read  by 
its  vernier  to  single  minutes. 

It  has  also  a  clamp-screw  inserted  near  its  centre,  by  which 
it  can  be  set  fast  to  the  telescope  axis  in  any  desired  position. 

The  vernier  of  the  vertical  limb  is  made  movable  by  the 
tangent-screw  attached,  so  that  its  zero  and  that  of  the  limb 
are  readily  made  to  coincide  when,  in  adjusting  the  limb  to  the 
level  of  the  telescope,  the  arc  is  clamped  to  the  axis. 

The  usual  tangent  movement  to  the  telescope  axis  serves,  of 
course,  to  bring  the  vertical  limb  to  the  proper  elevation,  as 
hereafter  described. 

A  level  on  the  under  side  of  the  telescope,  with  ground  vial 
and  scale,  is  indispensable  in  the  use  of  the  solar  attachment. 

The  divided  arcs,  verniers,  and  hour  circle,  are  all  on  silver 
plate,  and  are  thus  easily  read  and  preserved  from  tarnishing. 


THE  ADJUSTMENTS. 

132.  The  Solar  Lenses  and  Lines  are  adjusted  precisely  like 
those  of  the  ordinary  solar,  the  declination  arm  being  first  de- 
tached by  removing  the  clamp  and  tangent  screws,  and  the 
conical  centre  with  its  two  small  screws,  by  which  the  arm  is 
attached  to  the  arc. 

The  adjuster,  which  is  a  short  bar  f urn, shed  with  every 
instrument,  is  then  substituted  for  the  declination  arm,  the 
conical  centre  screwed  into  its  place  at  one  end,  and  the 
clamp-screw  into  the  other,  being  inserted  through  the  hole  left 
by  the  removal  of  the  tangent-screw,  thus  securing  the  adjuster 
firmly  to  the  arc. 

The  arm  is  then  turned  to  the  sun,  as  described  in  the  article 
on  the  Solar  Compass,  and  reversed  by  the  opposite  faces  of 
the  blocks  upon  the  adjuster,  until  the  image  will  remain  in  the 
centre  of  the  equatorial  lines.  This  adjustment  is  very  rarely 
needed,  as  the  lenses  are  cemented  in  their  cells,  and  the  plates 
securely  fastened. 


84  PLANE   SURVEYING. 

133.  The  Vernier  of  the  Declination  Arc  is  adjusted  by  set- 
ting the  vernier  at  zero,  and  then  raising  or  lowering  the  tele- 
scope by  the  tangent-screw,  until  the  sun's  image  appears  exactly 
between  the  equatorial  lines. 

Having  the  .telescope  axis  clamped  firmly,  carefully  revolve 
the  arm  until  the  image  appears  on  the  other  plate. 

If  preciseh'  between  the  lines,  the  adjustment  is  complete ; 
if  not,  move  the  declination  arm  by  its  tangent-screw,  until  the 
image  will  come  precisely  between  the  lines  on  the  two  opposite 
plates  ;  clamp  the  arm  and  remove  the  index  error  by  loosening 
two  flat-head  screws  on  the  back,  which  fasten  the  movable  arc 
to  the  declination  limb ;  place  the  zero  of  the  limb  and  vernier 
in  exact  coincidence  and  the  adjustment  is  finished. 

134.  To  Adjust  the  Polar  Axis.     First  level  the  instrument 
carefully  by  the  long  level  of  the  telescope,  using  in  the  opera- 
tion the  tangent  movement  of  the  telescope  axis  in  connection 
with  the  levelling  screws  of  the  parallel  plates,  until  the  bubble 
will  remain  in  the  centre  during  a  complete  revolution  of  the  in- 
strument upon  its  axis. 

Place  the  equatorial  sights  on  the  top  of  the  blocks  as  closely 
as  is  practicable  with  the  distinct  view  of  a  distant  object ;  and 
having  previously  set  the  declination  arm  at  zero,  sight  through 
the  interval  between  the  equatorial  sights  and  the  blocks  at  some 
definite  point  or  object,  the  declination  arm  being  placed  over 
either  pair  of  the  capstan-head  screws  on  the  under  side  of  the 
disc. 

Keeping  the  declination  arm  upon  the  object  with  one  hand, 
with  the  other  turn  the  instrument  half  around  on  its  axis,  and 
sight  upon  the  same  object  as  before.  If  the  sight  strikes 
either  above  or  below,  move  the  two  capstan-head  screws  imme- 
diately under  the  arm,  loosening  one  and  tightening  the  other 
as  may  be  needed,  until  half  the  error  is  removed. 

Sight  again  and  repeat  the  operation,  if  needed,  until  the 
sight  will  strike  the  same  object  in  both  positions  of  the  instru- 
ment, when  the  adjustment  of  the  axis  in  one  direction  will  be 
complete. 


DESCRIPTION   OF   INSTRUMENTS.  85 

Now  turn  the  instrument  at  right  angles,  keeping  the  sight 
still  upon  the  same  object  as  before  ;  if  it  strikes  the  same  point 
when  sighted  through,  the  axis  will  be  truly  vertical  in  the  sec- 
ond position  of  the  instrument. 

If  not,  bring  the  sight  upon  the  same  point  by  the  other  pair 
of  capstan-head  screws  now  under  the  declination  arc,  reverse 
as  before,  and  continue  the  operation  until  the  same  object  will 
keep  in  the  sight  in  all  positions,  when  the  polar  axis  will  be 
made  precisely  at  right  angles  to  the  level  and  to  the  line  of 
collimation  of  the  transit. 

It  should  here  be  noted  that  as  this  is  by  far  the  most  delicate 
and  important  adjustment  of  the  solar  attachment,  it  should  be 
made  with  the  greatest  care,  the  bubble  kept  perfectly  in  the 
centre  and  frequently  inspected  in  the  course  of  the  operation. 

135.  To  Adjust  the  Hour  Arc.    Whenever  the  instrument  is 
set  in  the  meridian,  as  will  be  hereafter  described,  the  index  of 
the  hour  arc  should  read  apparent  time. 

If  not,  loosen  the  two  flat-head  screws  on  the  top  of  the  hour 
circle,  and  with  the  hand  turn  the  circle  around  until  it  does, 
fasten  the  screws  again,  and  the  adjustment  will  be  complete. 

To  obtain  mean  time,  of  course  the  correction  of  the  equa- 
tion for  the  given  day,  as  given  in  the  Nautical  Almanac,  must 
always  be  applied. 

136.  To  Find  the  Latitude.     First  level  the  instrument  very 
carefully,  using,  as  before,  the  level  of  the  telescope  until  the 
bubble  will  remain  in  the  centre  during  a  complete  revolution 
of  the  instrument,  the  tangent  movement  of  the  telescope  being 
used  in  connection  with  the  levelling  screws  of  the  parallel  plates, 
and  the  axis  of  the  telescope  firmly  clamped. 

Next  clamp  the  vertical  arc  so  that  its  zero  and  that  of  its 
vernier  coincide  as  near  as  may  be,  and  then  bring  them  into 
exact  line  by  the  tangent-screw  of  the  vernier. 

Then,  having  the  declination  of  the  sun  for  12  o'clock  of  the 
given  day  as  affected  by  the  meridional  refraction  carefully  set 


86  PLANE   SURVEYING. 

off  upon  the  declination  arc,  note  also  the  equation  of  time  and 
fifteen  or  twenty  minutes  before  noon,  the  telescope  being 
directed  to  the  north,  and  the  object-end  lowered  until,  by 
moving  the  instrument  upon  its  spindle  and  the  declination  arc 
from  side  to  side,  the  sun's  image  is  brought  nearly  into  posi- 
tion between  the  equatorial  lines.  Now  bring  the  declination 
arc  directly  in  line  with  the  telescope,  clamp  the  axis  firmly, 
and  with  the  tangent-screw  bring  the  image  precisely  between 
the  lines  and  keep  it  there  with  the  tangent-screw,  raising  it  as 
long  as  it  runs  below  the  lower  equatorial  line,  or,  in  other 
words,  as  long  as  the  sun  continues  to  rise  in  the  heavens. 

When  the  sun  reaches  the  meridian  the  image  will  remain 
stationary  for  an  instant,  and  then  begin  to  rise  on  the  plate. 

The  moment  the  image  ceases  to  run  below  is  of  course  ap- 
parent noon,  when  the  index  of  the  hour  arc  should  indicate 
XII,  and  the  latitude  be  determined  by  the  reading  of  the  ver- 
tical arc. 

It  must  be  remembered,  however,  that  the  angle  through 
which  the  polar  axis  has  moved  in  the  operation  just  described 
is  measured  from  the  zenith  instead  of  the  horizon,  as  in  the 
ordinary  solar,  so  that  the  angle  read  on  the  vertical  limb  is  the 
complement  of  the  latitude. 

The  latitude  itself  is  readily  found  by  subtracting  this  angle 
from  90° ;  thus  at  Troy,  the  reading  of  the  limb  being  found  as 
above  directed  to  be  47°  16',  the  latitude  will  be 

90° -47°  16' =  42°  44'. 

It  will  be  noticed  that  with  this  apparatus  the  latitude  of  any 
place  can  be  most  easily  ascertained  without  any  index  error, 
as  in  the  usual  solar  compass. 

137,  To  Use  the  Solar  Attachment.  From  the  foregoing  de- 
scription it  will  be  readily  understood  that  good  results  cannot 
be  obtained  from  the  solar  attachment  unless  the  transit  is  of 
good  construction, — furnished  with  the  appliances  of  a  level  on 
telescope,  clamp  and  tangent  movement  to  axis,  and  vertical 


DESCRIPTION   OF   INSTRUMENTS.  87 

arc  with  adjustable  vernier,  and  the  sockets  or  centres  in  such 
condition  that  the  level  of  the  telescope  will  remain  in  the  cen- 
tre when  the  instrument  is  revolved  upon  either  socket. 

138.  To  Bun  Lines  with  the  Solar  Attachment.  Having  set 
off  the  complement  of  the  latitude  of  the  place  on  the  vertical 
arc,  and  the  declination  for  the  given  day  and  hour  as  in  the 
solar,  the  instrument  being  also  carefully  levelled  by  the  tele- 
scope bubble,  set  the  horizontal  limb  at  zero,  and  clamp  the 
plates  together,  loosen  the  lower  clamp  so  that  the  transit 
moves  easily  upon  its  lower  socket,  set  the  instrument  approxi- 
mately north  and  south,  the  object-end  of  the  telescope  point- 
ing to  the  north,  turn  the  proper  solar  lens  to  the  sun,  and, 
with  one  hand  on  the  plates  and  the  other  on  the  revolving  arm, 
move  them  from  side  to  side,  until  the  sun's  image  is  brought 
between  the  equatorial  lines  on  the  silver  plate. 

The  lower  clamp  of  the  instrument  should  now  be  fastened, 
and  any  further  lateral  movement  be  made  by  the  tangent- 
screw  of  the  tripod.  The  necessary  allowance  being  made  for 
refraction,  the  telescope  will  be  in  the  true  meridian,  and  being 
undamped,  may  be  used  like  the  sights  of  the  ordinary  solar 
compass,  but  with  far  greater  accuracy  and  satisfaction  in 
establishing  meridian  lines.  Of  course  when  the  upper  or 
vernier  plate  is  undamped  from  the  limb,  any  angle  read  by  the 
verniers  is  an  angle  from  the  meridian,  and  thus  parallels  of 
latitude  or  any  other  angles  from  the  true  meridian  may  be 
established  as  with  the  solar  compass. 

The  bearing  of  the  needle,  when  the  telescope  is  on  the  meri- 
dian, will  also  give  the  variation  of  the  needle  at  the  point  of 
observation. 

The  declination  of  the  needle  being  set  off,  and  the  needle 
kept  then  at  zero,  or  "  with  the  sun,"  lines  may  be  run  by  the 
needle  alone  when  the  sun  is  obscured. 

Though  when  not  inconsistent  with  the  remarks  following  the 
table  on  page  95,  the  sun  should  be  observed  for  direction  at 
every  station. 


88  PLANE  SURVEYING. 


THE  SAEGMULLER  ATTACHMENT. 

139.  As  seen   in   the   engraving   on  the   opposite  page,  it 
consists  essentially  of  a  small  telescope  and  level,  the  telescope 
being  mounted  in  standards,  in  which  it  can  be  elevated  or 
depressed.     The  standards  revolve  around  an  axis,  called  the 
polar  axis,  which  is  fastened  to  the  telescope  axis  of  the  transit 
instrument.     The  telescope,  called  the  "  Solar  Telescope,"  can 
thus  be  moved  in  altitude  and  azimuth.    Two  pointers,  attached 
to  the  solar  telescope  to  approximately  set  the  instrument,  are 
so  adjusted  that  when  the  shadow  of  the  one  is  thrown  upon 
the  other  the  sun  will  appear  in  the  field  of  view. 

140.  Adjustments.    When  the  apparatus  is  attached  to  the 
transit,  which  instrument  must  be  in  good  adjustment,  its  polar 
axis  should  be  at  right  angles  both  to  the  horizontal  axis  of  the 
main  telescope  and  to  the  line  of  collimation. 

TEST.  Level  the  transit,  and  bring  the  bubble  of  each  tele- 
scope to  the  centre  of  its  run.  Revolve  the  solar  telescope 
about  its  polar  axis,  and  if  its  bubble  remains  central,  this  ad- 
justment is  complete.  If  not,  correct  half  the  movement  by 
the  adjusting  screws  at  the  base  of  the  polar  axis,  and  the  other 
by  revolving  the  solar  telescope  on  its  horizontal  axis. 

141.  Second.     The  line  of  collimation  of  the  solar  telescope 
and  the  axis  of  its  attached  level  must  be  parallel. 

TEST.  Bring  the  telescopes  into  the  same  vertical  plane,  and 
the  large  bubble  to  the  middle  of  its  run.  Direct  then  the  tran- 
sit telescope  to  a  mark  at  a  convenient  distance  away,  say  100 
feet;  point  also  the  "  solar"  to  a  mark  above  this  equal  to  the 
distance  between  their  axes.  If  now  the  bubble  of  the  solar 
telescope  is  not  in  the  middle  of  the  tube,  make  it  so  by  the 
adjusting  screws,  and  the  instrument  will  be  in  adjustment. 

When  the  combined  instrument  is  in  proper  adjustment  the 
bubbles  of  the  telescopes  and  plates  will  be  in  the  middle  of 
their  tubes,  and  the  lines  of  collimation  parallel. 


TRANSIT  WITH   SOLAR   ATTACHMENT, 

AS    MADE    BY   FAUTH    &  C'<).,  \VASUINOTON,   D.C. 


DESCRIPTION  OF   INSTRUMENTS.  91 

All  the  adjustments,  including  those  of  the  transit,  should  be 
frequently  examined,  and  kept  as  nearly  perfect  as  possible. 

142.  The  advantages  of  solar  attachments  over  the  ordinar}7 
solar  compass  consist  principally  in  the  telescopic  sight,  and  the 
use  of  a  vertical  limb  to  set  off  declination  and  co-latitude. 

LATITUDE. 
By  the  Sun.  —  With  Saegmuller's  Attachment. 

143.  Level  the  transit  carefully,  point  the  telescope  south, 
and  elevate  or  depress  the  object-end,  according  as  the  decli- 
nation of  the  sun  is  south  or  north,  an  amount  equal  to  the 
declination.*     Bring  the  solar  telescope  into  the  vertical  plane 
of  the  main  telescope,  level  it  carefully,  and  clamp  it.     With 
the  solar  telescope  observe  the  sun  a  few  minutes  before  his 
culmination,   bring  the   horizontal  middle  wire  tangent  to  the 
upper  limb  by  moving  the  transit   telescope   in   altitude   and 
azimuth,  and  keep  it  so  by  the  slow-motion  screws  until  the  sun 
ceases  to  rise.     Then  take  the  reading  of  the  vertical  arc,  cor- 
rect for  index  error,  if  any,  for  refraction  due  to  altitude,!  as 
per  table  below  ;  diminish  the  result  by  the  sun's  semi-diameter, 
and  subtract  the  result  from  90°  for  the  latitude. 


*  For  declination,  consult  a  nautical  almanac. 

t  Corrected  for  index  error,  the  arc  reading  would  be  the  sum  of  the 
co-latitude  and  refraction.  The  refraction  being  due  to  the  meridian  alti- 
tude of  the  sun,  which  altitude  in  the  United  States  is  equal  to  the  alge- 
braic sum  of  the  declination  and  co-latitude. 


92 


PLANE   SURVEYING. 


TABLE  OF  MEAN   REFRACTIONS   OF   CELESTIAL   OBJECTS   FOR  TEMPER- 
ATURE 50°,  AND  BAROMETER  29.6  INCHES. 


ALTITUDE. 

REFRACTION. 

ALTITUDE. 

REFRACTION. 

10° 

5'  15" 

20° 

2'  35" 

11 

4  47 

25 

2  02 

12 

4  23 

30 

1  38 

13 

4  03 

35 

1  21 

14 

3  45 

40 

1  08 

15 

3  30 

45 

0  57 

16 

3  17 

50 

0  48 

17 

3  04 

60 

0  33 

18 

2  54 

70 

0  21 

19 

2  44 

80 

0  10 

By  interpolation,  the  refraction,  due  to  any  altitude  within  the 
limits  of  the  table  may  be  found. 


LATITUDE  BY  CIRCUMPOLAR  STAR. 

144.  The  arc  measuring  the  angle  of  elevation  of  the  pole  at  any 
station  indicates  the  latitude  of  that  station.  If,  then,  the  place  of  the  pole 
were  indicated  by  a  heavenly  body,  its  altitude  measured  and  corrected 
for  refraction  would  give  at  once  the  latitude. 

There  being  no  such  body,  a  circumpolar  star  may  be  used.  Take  its 
altitude  at  either  culmination,  subtract  refraction  due  to  altitude,  and  the 
remainder,  increased  or  diminished  by  the  polar  distance  according  as  the 
lower  or  upper  culmination  was  observed,  will  give  the  latitude. 

Better,  when  practicable,  to  observe  both  culminations,  correct  for  re- 
fraction, and  take  the  arithmetical  mean  of  the  result.  Still  greater  accu- 
racy would  be  obtained  by  taking  the  mean  of  observations  at  upper  and 
lower  transit  of  several  eircumpolar  stars. 

If  A  and  A'  respectively  denote  the  angles  measuring,  from  the  north, 
the  altitudes  of  a  circumpolar  star  at  its  upper  and  lower  culminations, 
and  r  and  r'  the  corresponding  refractions,  then, 


latitude  = 


—  (r  +  r')]. 


DESCRIPTION    OF   INSTRUMENTS.  93 


To  FIND  THE   MERIDIAN  AND   DECLINATION  OF  THE  NEEDLE, 
USING  THE  ATTACHMENT.* 

145.  First.  Take  the  declination  of  the  sun  as  given  in  the 
Nautical  Almanac  for  the  given  day,  and  correct  it  for  retrac- 
tion and  hourly  change.  Incline  the  transit  telescope  until  this 
amount  is  indicated  by  its  vertical  arc.  If  the  declination  of 
the  sun  is  north,  depress  the  object-end ;  if  south,  elevate  it. 
Without  disturbing  the  position  of  the  transit  telescope,  bring 
the  solar  telescope  into  the  same  vertical  plane,  and  make  it 
horizontal  by  means  of  its  level.  The  two  telescopes  will  then 
form  an  angle  which  equals  the  amount  of  the  declination,  and 
the  inclination  of  the  solar  telescope  to  its  polar  axis  will  be 
equal  to  the  polar  distance  of  the  sun. 

Second.  Without  disturbing  the  relative  positions  of  the  two 
telescopes,  incline  them  and  set  the  vernier  to  the  co-latitude  of 
the  place. 

By  moving  the  transit  and  the  solar  attachment  around  their 
respective  vertical  axes,  the  image  of  the  sun  will  be  brought 
into  the  field  of  the  solar  telescope,  and  after  accurately  bisecting 
it  the  transit  telescope  must  be  in  the  meridian,  and  the  compass- 
needle  indicates  its  deviation  at  that  place. 

The  vertical  axis  of  the  solar  attachment  will  then  point  to 
the  pole,  the  apparatus  being  in  fact  a  small  equatorial.  Re- 
volve the  main  telescope  on  its  horizontal  axis,  and  set  a  mark 
at  a  convenient  distance, — 1000  feet  if  practicable. 

Make  a  reverse  observation  as  follows  :  Turn  the  transit  180* 
in  azimuth,  and  set  off  the  declination,  elevating  or  depressing 
now  the  eye-end,  according  as  the  declination  is  south  or  north; 
bring  the  object-end  of  the  solar  telescope  to  point  in  the  direc- 
tion of  the  eye-end  of  that  of  the  main  instrument,  and  level  it. 
Set  the  vertical  arc  to  the  co-latitude  of  the  place,  and  complete 
the  observation  as  before.  Reverse  the  large  telescope  on  its 

*  For  other  methods,  see  Chapter  III.,  p.  218,  and  Chapter  VI.,  Solar 
Compass. 


94  PLANE   SURVEYING. 

horizontal  axis,  and  see  if  it  points  to  the  mark  set  by  the  direct 
observation ;  if  it  do  not,  take  the  mean  of  the  two  pointings 
for  the  meridian. 

If  greater  accuracy  is  required,  make  other  observations  at 
different  hours  of  the  day,  under  different  conditions  of  the 
atmosphere,  and  compare  results  with  those  given  in  Chapters 
III.  and  VI. 

146.  Time  and  azimuth  are  calculated  from 
an  observed  altitude  of  the  sun  by  solving  the 
spherical  triangle  formed  by  the  sun,  the  pole, 
and  the  zenith  of  the  place.  The  three  sides, 
SP,  PZ,  ZS.  complements  respectively  of  the 
declination,  latitude,  and  altitude  are  given, 
and  we  hence  deduce  SPZ,  the  hour  angle, 
from  apparent  noon,  and  PZS  the  azimuth 
of  the  sun.* 

The  "  Solar  Attachment  "  solves  the  same 
spherical  triangle  by  construction,  for  the 
second  process  brings  the  vertical  axis  of  the  solar  telescope  to 
the  required  distance  ZP  from  the  zenith,  while  the  first  brings 
it  to  the  required  distance  SP  from  the  sun. 

If  the  two  telescopes,  both  being  in  position  —  one  in  the 
meridian,  and  the  other  pointing  to  the  sun  —  are  now  turned 
on  their  horizontal  axes,  the  vertical  remaining  undisturbed, 
until  each  is  level,  the  angle  between  their  directions  —  found 
by  sighting  on  a  distant  object  —  is  SPZ,  the  time  from  appar- 
ent noon. 

This  gives  an  easy  observation  for  correction  of  time-piece. 

147.  An  error  either  in  the  declination  or  latitude  will  cause 
an.  error  in  the  azimuth. 

These  errors  in  azimuth  corresponding  to  one-minute  error  in 
declination  or  latitude,  for  various  hours  and  half-hours  of  the 

*  A  Table  of  Equation  of  Time  is  given  at  the  end  of  this  book  which 
will  be  useful  in  solving  analytically  the  spherical  triangle  PZS  for  time. 


DESCRIPTION    OF   INSTRUMENTS.  95 

day,  and  for  different   latitudes,   have   been  computed   and 
tabulated.* 


THE  SAEGMULLEB  ATTACHMENT. 

The  table  indicates  the  best  time  to  observe  the  sun  for  merid- 
ian, or  to  determine  the  true  bearing  of  a  line,  to  be  soon  after 
sunrise  or  just  before  sunset. 

However,  on  account  of  refraction  at  these  times  being  great 
and  very  uncertain,  it  is  best  in  general  not  to  make  the  obser- 
vation when  the  sun  is  nearer  the  horizon  than  about  15  degrees. 
Moreover,  the  solar  apparatus  should  not  be  relied  on  for  very 
accurate  work  between  10  A.M.  and  2  P.M. 

An  error  in  latitude  does  not  cause  an  error  in  azimuth  when 
the  sun  is  in  the  pole  of  the  meridian. 

148.  The  Stadia,  or  Micrometer,  is  a  compound  cross-wire 
ring  or  diaphragm,  shown  below,  having  three  horizontal  wires, 
of  which  the  middle  one  is  cemented  to  the  ring  as  usual,  while 
the  others,  bb  and  cc,  are  fastened  to  small  slides,  held  apart  by 

*  By  Professor  H.  T.  Stewart,  C.  E.,  of  the  Western  University  of 
Pennsylvania. 


96 


PLANE   SURVEYING. 


a  slender  brass  spring  hoop,  and  actuated  by  independent  screws 
del,  by  which  the  distance  between  the  two  movable  wires  can  be 
adjusted  to  include  a  given  space ;  as,  1  foot  on  a  rod  100  feet 
distant.  These  wires  will  in  the  same  manner  include  2  feet  on 


a  rod  200  feet  distant,  or  half  a  foot  at  a  distance  of  50  feet, 
and  so  on  in  the  same  proportion  ;  thus  furnishing  a  means  of 
measuring  distances  —  especially  over  broken  ground  —  much 
more  easily,  and  even  more  accurately,  than  with  a  tape  or 
chain. 

149.    Its  principles  may  be  explained  more  fully  as  follows : 


Let  the  above  figure  represent  a  section  of  a  common  tele- 
scope with  but  two  lenses,  between  which  the  diaphragm  with 
the  stadia  wires  is  placed,  and  assume  that 

/  =  the  focal  distance  of  the  object-glass  ; 

p  =  the  distance  of  the  stadia  wires  a  and  b  from  each  other ; 

d  =  the  horizontal  distance  of  the  object-glass  to  the  stadia  . 


DESCRIPTION   OF  INSTRUMENTS.  97 

a  =  stadia  reading  (BA)  ; 

D  =  horizontal  distance  from  middle  of  instrument  to  stadia. 

The  telescope  is  levelled  and  sighted  to  a  levelling  or  stadia 
rod,  which  is  held  vertically,  hence  at  right  angles  with  the  line 
of  sight.  According  to  a  principle  of  optics,  rays  parallel  to 
the  axis  of  the  lens  meet,  after  being  refracted,  in  the  focus  of 
the  lens.  Suppose  the  two  stadia  wires  are  the  sources  of  those 
rays,  we  have,  from  the  similarity  of  the  two  triangles  a'b'F 
and  FAB,  the  proportion 

d  -f:a=f:  p. 

The  quotient  f:p  is,  or  at  least  can  be  made,  constant,  and 
may  be  designated  by  k ;  hence  we  may  write 

d-f=FC  =  ka.    * 

To  get  the  distance  from  the  centre  N  of  the  instrument  there 
must  be  added  to  FC  the  value 

c  =  OF+ON. 

ON  is  mostly  equal  to  half  the  focal  length  of  the  object- 
glass ;  hence, 

c=1.5/. 

Therefore  the  formula  for  the  distance  of  the  stadia  from  the 
centre  of  instrument,  when  that  stadia  is  at  right  angles  to  the 
level  line  of  sight,  is 

D  =  ka  +  c.  (1) 

150.  When  the  line  of  sight  is  not  level,  it  is  impracticable, 
especially  in  long  distances,  to  hold  the  rod  in  a  vertical  plane, 
and  at  the  same  time  perpendicular  to  the  line  of  sight ;  hence 
it  is  customary  to  hold  the  rod  vertical,  as  in  the  preceding  case, 
and  obtain  the  true  distance  by  applying  a  correction  depending 
upon  the  angle  of  inclination  of  the  sight. 

This  correction  is  deduced  as  follows : 
Let    A  GB  =  2  w ; 

n  =  the  angle  of  inclination  ; 


98 


PLANE   SURVEYING. 


MF=  c  +  GF  =  c  +  k  x  CD  =  D' ; 

CD  must  be  expressed  by  AB ; 

MP=  the  horizontal  distance  =  D'  cos  « 


-p 


Now  the  angle 


or, 


Hence,        4? 


sin  m 


AF  = 


sin[90  +  (n- 
GF  sin  m 


and  Z* 


But 

and 


cos  (w  —  m) 
BF  sin  m 


sin  [90  —  (w 
(rF  sin  m 


cos  (n  -f  m) 

r          I 

=  GFs\nm\ |- 

[_cos  (n  —  m) 

_     CD    _  CD  cos  m 
2tanm~  2sinm. 


DESCRIPTION    OF    INSTRUMENTS.  99 

Substituting  this  value  of  GF  in  the  equation  above,  we 
obtain 

^_CD  cos  m  [cos  (n  +  m)+  cos  (n  —  m)] 

2  cos  (n  +  m)  cos  (n  —  m) 
^r,         cos2n  cos2w  —  sin2?*  sin2m 

Ol  ,  Ui/  =  U  •  -    -  7 

cos  n  cos"  w 
and          Z>'  =  c  +  &o  c088  M 


. 

cosw  cos2m 
Whence, 

D  —  c  cos  ?i  -f-  fra  cos2w  —  fca  sin2w  tan2ra. 

The  third  term  of  second  member  of  this  equation  may  be 
neglected,  as  it  is  very  small,  even  for  long  distances  and  large 
angles  of  elevation  (for  1500',  n  =  45°  and  k  =  100,  it  is  but 
0.07')  ;  therefore  the  final  formula  for  distances,  with  a  stadia 
kept  vertical,  and  with  wires  equidistant  from  the  centre  wire, 
is  the  following: 

D  =  c  cos  n  +  ok  cos2w.  (2) 

The  value  of  ccosw  is  usually  neglected,  as  it  amounts  to  but 
1  or  1.5  feet  ;  it  is  exact  enough  to  add  always  1.25'  to  the  dis- 
tance as  derived  from  the  formula 


(2o)* 

151.  The  focal  length  /of  the  object-glass  may  be  found  by 
focussing  the  instrument  upon  some  distant  object,  say  a 
heavenly  body,  and  measuring  then  the  distance  between  the 
plane  of  the  cross-wires  and  that  of  the  objective.  ON,  being 
equal  to  the  distance  between  the  objective  and  the  intersection 
of  a  plumb-line  with  the  horizontal  axis  of  the  telescope,  may 
be  obtained  by  direct  measurement. 

The  distance  p,  between  the  stadia  wires,  may  be  determined 
as  follows  : 

Set  up  the  instrument  on  level  ground,  or  nearly  so,  and 
measure  forward  from  the  plumb-line  a  distance  equal  to  c,  and 

*  The  above  explanation  of  the  stadia  is  substantially  that  given  by 
Mr.  G.  J.  Specht,  published  by  Van  Nostrand,  1884,  though  corrected  and 
simplified. 


100  /        PLANE    SURVEYING. 

mark  the  point ;  measure  onward  from  the  mark  any  convenient 
distance  d,  400  or  500  feet,  as  a  base.  The  telescope  being 
level,  observe  carefully  the  space  a  intercepted  by  the  stadia 
wires  on  a  levelling-rod  held  vertically  at  the  farther  extremity 
of  the  base. 

Then  from  the  proportion  d  —f:  a  =f :  p  the  required  dis- 
tance p  may  be  obtained. 

EXAMPLES. 

1.  Given /=  8  inches,  base  =  500  feet,  and  a  =  5. 25   feet. 
Find p=. 084  inches. 

2.  At   what   fractional   part   of  the  focal  length  must  the 
stadia  wires  be  separated  so  that  one  foot  on  the  rod  will  cor- 
respond to  100  feet  base?     State  also  the  distance  between  the 
wires  in  terms  of  the  focal  length,  when  one  foot  on  rod  cor- 
responds to  66  feet  base. 

3.  Measure  with  a  stadia  one  or  more  sides  of  a  field,  also 
the  distance  across  a  valley,  or  from  one  ridge  to  another,  and 
compare  the  results  with  chain  measurement  between  the  same 
points. 

4.  Measure  with  the  stadia  up  or  down  a  hillside,  and  chain 
between  the  same  points.     Compare  results. 

GRADIENTER. 

152.  This  attachment,  as  shown  on  next  page,  is  often  used 
with  transits  for  fixing  grades,  determining  distances,  etc. 

It  consists  mainly  of  a  screw  attached  to  the  semicircular 
expanded  arm  of  the  ordinary  clamp  of  the  telescope  axis  ;  the 
screw  is  accurately  cut  to  a  given  number  of  threads,  and  pass- 
ing through  a  nut  in  one  side  of  the  arm,  presses  against  a  little 
stud  A  fixed  to  the  inside  surface  of  the  right-hand  standard. 

In  the  other  side  of  the  semicircular  arm  is  inserted  a  hollow 
cylinder  containing  a  pin  actuated  by  a  strong  spiral  spring,  the 
end  of  the  pin  pressing  against  the  side  of  the  stud  opposite 
that  in  contact  with  the  screw. 


DESCRIPTION   OF   INSTRUMENTS.  101 

Near  the  other  end  of  the  screw,  and  turning  with  it,  is  a 
wheel,  or  micrometer,  the  rim  of  which  is  plated  with  silver, 
and  divided  into  one  hundred  equal  parts. 

A  small  silver  scale,  attached  to  the  arm  and  just  above  the 
micrometer  wheel,  is  divided  into  spaces,  each  of  which  is  just 
equal  to  one  revolution  of  the  screw ;  so  that  by  comparing  the 
edge  of  the  wheel  with  the  divisions  of  the  scale,  the  number  of 
complete  revolutions  of  the  screw  can  be  easily  counted. 


It  will  be  seen  that  when  the  clamp  is  made  fast  to  the  axis 
by  the  clamp-screw,  and  the  gradienter-screw  turned,  it  will 
move  the  telescope  vertically,  precisely  like  the  tangent-screw 
ordinarily  used. 

And  as  the  value  of  a  thread  is  such  that  a  complete  revolu- 
tion of  the  screw  will  move  the  horizontal  cross-wire  of  the 
telescope  over  a  space  of  one  foot  on  a  rod  at  a  distance  of 
one  hundred  feet,  it  is  clear  that  when  the  screw  is  turned 
through  fifty  spaces  on  the  graduated  head,  the  wire  will  pass 
over  fifty  one-hundredths,  or  one-half  a  foot  on  the  rod,  and  so 
on  in  the  same  proportion. 


102 


PLANE   SURVEYING. 


In  this  way  the  gradienter  can  be  used  in  the  measurement 
of  distances,  precisely  like  the  stadia  just  described. 

Grades  can  also  be  established,  with  great  facility,  as  fol- 
lows :  First,  level  the  instrument ;  bring  the  telescope  level  to 
its  centre  by  the  clamp  and  gradienter  screw  ;  move  the  gradu- 
ated head  until  its  zero  is  brought  to  the  edge  of  the  scale  ;  and 
then  turn  off  as  many  spaces  on  the  head  as  there  are  hun- 
dredths  of  feet  to  the  hundred  in  the  grade  to  be  established. 


SECTION  II. 

A.    BEARINGS  WITH  COMPASS. 

153.  To  Obtain  the  Bearing  of  a  Line.  At  one  end  of  the 
line,  or  at  any  other  point  in  it,  set  up  and  level  the  compass, 
loosen  the  needle,  and  direct  the  sights  toward  the  other  end. 
The  degree  on  which  the  needle  comes  to  rest  will  indicate  the 
angle  between  the  magnetic  meridian  and  the  direction  of  the 
line,  or  the  bearing. 

For  example,  if  the  line  lies  between  the  north  and  east  points, 
as  OP,  and  the  angle  NOP  being,  say  42  degrees,  the  bearing 
of  the  line  OP\s  written,  N.  42°  E., 
and  read,  "north  forty-two  degrees 
east."  If,  as  OP',  it  lies  between 
south  and  east,  and  the  angle  SOP' 
is,  say  74  degrees,  it  is  written, 
S.  74°  E.,  and  read,  "south  sev- 
enty-four degrees  east" ;  in  like 
manner  for  lines  in  other  quadrants. 
It  will  be  observed  that  the  bear- 
ing of  a  line  does  not  exceed  90°. 
A  line  which  might  be  read  "  N. 

90°  W."  or  "  S.  90°  W."  is  recorded  as  west.  The  bearing  can 
be  read  most  accurately  by  placing  the  eye  over  one  end  of  the 
needle  and  taking  the  reading  from  the  other  end. 


BEARINGS    WITH   COMPASS. 


103 


Since  the  graduations  are  usually  made  to  half -degrees,  the 
bearing  can  be  taken  quite  accurately  to  quarter-degrees,  and 
by  practice,  even  closer,  without  the  use  of  the  vernier.  In 
fact,  the  principal  use  of  the  vernier  on  a  compass  is  to  facili- 
tate the  running  of  lines  from  old  deeds,  where,  when  the 
declination  is  ascertained,  it  is  turned  off  on  the  vernier,  and 
the  surveyor  may  use  then  the  bearings  as  given  in  the  deed  by 
which  he  is  surveying  the  tract,  without  making  a  calculation 
for  the  bearing  of  each  line.  The  vernier  cannot  be  relied  on 
to  read  bearings  to  minutes,  on  account  of  the  difficulty  of 
accurately  manipulating  it. 

154.  Reverse  Bearings.   Since  in  plane  surveying  the  meri- 
dians passing  through  the  extremities  of  a  line  are  considered 
parallel,  the  direct  and  reverse  bearings  should  indicate  the  same 
angle.    That  is  to  say,  a  line,  as  LM,  the  bearing  of  which,  taken 
ati,  called  also  fore-sight,  is  N.  40°  E.,  when 

taken  at  M,  back-sight,  should  be  S.  40°  W.  ; 
the  degrees  being  the  same,  the  letters  indi- 
cating the  opposite  cardinal  points. 

When  surveying  a  tract  of  land  with  the 
compass,  the  instrument  should  be  set  up  at 
every  corner,  and  the  bearing  and  reverse  £, 
bearing  of  every  line  taken,  as  a  check  on  the 
observer's  reading  and  the  working  of  the 
needle,  since  a  disagreement  in  the  angle  thus 
measured  would  be  evidence  sufficient  to  war- 
rant a  review  of  the  work. 

155.  Local  Attraction.    If  the  readings  of  the  needle  of  the 
fore-sight  and  back-sight  have  been  correctly  made,  and  there 
is  found  a  disagreement,  local  attraction  exists.     It  is  usually 
caused  by  the  presence  of  ferruginous  matter.     It  may  exist  at 
both  stations  or  at  only  one  of  them. 

Assuming  that  the  direct  and  reverse  bearings  of  the  preced- 
ing line  agree,  then  the  difference  in  the  reading  at  the  two  ends 


104  PLANE   SURVEYING. 

of  the  line,  when  the  attraction  exists,  will  show  the  local  vari- 
ation at  the  last  station,  and  this  correction  must  be  applied  to 
the  reading  of  the  needle  for  the  bearing  of  the  next  line.  If, 
however,  the  needle  will  not  reverse  on  the  first  line  of  a  survey, 
then  it  will  be  necessary  to  set  up  at  some  other  point  of  the 
tract ;  or,  if  this  is  impracticable,  select  one  or  more  stations 
near  the  suspected  points,  and  by  taking  the  bearings  of  these 
from  the  stations,  and  also  the  reverse  bearings,  the  intensity 
and  position  of  the  attraction  may  be  determined. 

156.  Proof  Bearings  and  Tests  of  Accuracy.  In  any 
important  compass  survey  it  is  well  to  check  the  work  by 
sighting  to  distant  prominent  objects,  such  as  buildings,  trees, 
etc.,  and  noting  the  readings.  Since  two  bearings  are  required 
to  locate  each  object,  —  and  until  it  is  located  it  cannot  serve  as 
a  check, — it  will  be  necessary  to  take  at  least  three  bearings  to 
each.  If,  then,  when  plotting,  the  three  lines  intersect  in  a 
point,  a  proof  is  given  of  the  correctness  of  the  measurements 
thus  connected.  The  lengths  and  bearings  of  diagonals  of  the 
tract  may  likewise  be  taken  as  checks  on  the  accuracy  of  the 
work ;  also,  when  in  plotting,  if  the  last  bearing  and  distance 
close  the  survey,  it  is  considered  a  proof  of  the  work.  The 
best  test,  however,  of  the  accuracy  of  the  survey  is  by  Latitudes 
and  Departures,  which  is  explained  in  Section  VI.  Articles  207 
and  208. 

It  may  be  well  to  caution  the  student  against  the  fallacy  of 
a  test  sometimes  given,  —  that  if  the  sum  of  the  interior  angles, 
determined  from  the  bearings,  equals  twice  as  many  right 
angles,  less  four,  as  the  figure  has  sides,  it  proves  the  work. 
This  "test,"  while  it  furnishes  proof  for  a  transit  survey  in 
which  the  interior  angles  have  been  measured,  will  not  show 
that  the  bearings  of  a  tract  have  been  correctly  taken.  The 
student  will  readily  perceive  the  truth  of  this  statement  if  he 
makes  or  imagines  a  plot  of  a  field  with  a  certain  side  the 
meridian,  then  conceives  the  whole  plot  turned  around  so  that 
another  side  comes  to  the  meridian,  it  will  be  evident  that 


BEARINGS   WITH   COMPASS.  105 

though  the  bearings  are  changed,  the  sum  of  the  interior  angles 
is  unaffected.  The  so-called  test  would  prove  the  work  in  either 
case. 

157.  Suggestions.  Test  frequently  to  see  that  the  instru- 
ment is  in  proper  adjustment.  Keep  the  same  end  ahead. 
Read  from  the  same  end  of  needle.  Sight  as  low  on  the  flag- 
staff as  possible.  Make  the  line  of  sight  as  nearly  horizontal 
as  practicable.  When  reading  near  the  cardinal  points,  be  care- 
ful that  the  bearing  is  not  read  in  the  wrong  quadrant,  also  that 
the  common  error  of  reading  56°  for  44°  is  not  committed.  See 
that  the  instrument  is  set  precisely  over 
the  station  from  which  the  measurements 
are  to  be  made  ;  that  the  flagstaff  is  exactly 
on  the  proper  point,  and  that  it  is  held 
plumb.  Level  the  instrument  carefully ; 
especially  see  that  it  is  level  across  the  line  of  sight.  Take 
the  bearing  and  measure  the  distance  on  the  true  line  when 
practicable ;  when  not,  because  of  a  high  fence,  bushes,  etc., 
set  off  the  least  perpendicular  distance  therefrom  at  both  ends 
which  will  afford  a  clear  view,  and  take  the  bearing  and  dis- 
tance of  the  extremities  of  these  perpendiculars. 

EXERCISES. 

1.  With  a  surveyor's  compass,  by  a  constant  and  direct  bear- 
ing only,  run  a  line,  say  40  chains  in  length,  over  hilly  ground, 
and  part  of  it,  if  possible,  through  brush ;  then  return,  using 
the  reverse  bearing  only. 

2.  With  the  same  instrument  run  another  line  equally  diffi- 
cult, using  both  direct  and  reverse  bearings  forward  and  back. 

3.  Make  a  survey  of  a  lot  one  side  of  which  is  near  to 
a  railroad  track.     If  local  attraction  is  found,  determine  its 
intensity. 

4.  Determine  the  magnetic  bearing  of  eacli  part  of  a  broken 
line  of  several  turns  along  a  railroad  track,  or  where  local 
attraction  is  known  to  exist. 


106  PLANE  SURVEYING. 


B.    ANGLES  WITH  TRANSIT. 

158.  With  the  Transit  the  survey  of  a  line  or  the  measure- 
ment of  an  angle  can  be  made  with  greater  accuracy  than  with 
the  compass,  since  the  reading  of  the  plates  to  minutes  sup- 
plants the  reading  of  the  needle   to   quarter  or  half-quarter 
degrees,  and  the  pointing  power  of  the  transit  greatly  exceeds 
that  of  the  compass. 

159.  To   measure  a  horizontal    angle,  as  MON.      Set  up 
the  instrument  precisely  at  0 ;  level  it  and  direct  the  intersec- 
tion of   the  wires  to  either  point,  say  N. 
Clamp  the  instrument  firmly  to  the  spindle, 
note  the  reading  of  the  vernier,  then  loosen 
the  vernier  plate  and  bring  the  telescope 
quite  near  the  other  line  so  that  its  ex- 
tremity M  is  in  the  field  of  view.     Clamp 
the  plate,  and  with   its   tangent   or   slow- 
motion  screw  bring  the  line  of  collimation 
precisely  on  M.     Again  take  the  reading. 

The  difference  of  the  two  readings  will  be  the  angle  required. 
It  is  more  convenient  to  make  the  first  sight,  ON,  with  the  zero 
of  the  limb  and  plate  coincident,  since  then  the  reading  of  the 
plates  after  observing  M  gives  at  once  the  angle.  If  at  each 
observation  but  one  vernier  is  read,  it  is  best  to  read  every 
time  from  the  same  one ;  it  is  better  at  each  observation, 
though,  to  read  both  verniers  and  take  the  mean  of  these, 
thereby  eliminating  eccentricit}*.  If,  however,  great  accuracy 
is  required,  the  measurement  of  the  angles  should  be  taken 
more  than  once,  by  the  method  of  repetition  or  by  series. 

160.  By  Repetition.  Make  an  observation  upon  any  point, 
and  read  both  verniers ;  clamp  the  lower  plate  tc  the  spindle, 
direct  the  telescope  to  another  point,  and,  as  a  check,  again 
read  the  verniers. 

Now,  keeping  the  index  at  the  last  reading,  turn  both  plates 


ANGLES    WITH   TRANSIT.  107 

back,  and  observe  again  on  the  first  point ;  clamp,  as  before, 
the  lower  plate,  and  turn  the  upper  one  so  as  to  sight  on  the 
second  point.  It  is  perceived  that  by  this  operation  the  angle 
has  been  measured  twice,  but  on  different  parts  of  the  limb. 
An  angle  may  obviously  be  repeated  any  number  of  times :  the 
mean  of  the  several  readings  gives  more  nearly  than  a  single 
measurement  the  true  angle.  The  reading  at  each  observation 
serves  as  a  check  on  the  work.  An  angle  may  be  repeated  by 
simply  noting  the  reading  at  the  first  and  last  observation, 
taking  their  difference,  and  dividing  by  the  number  of  repeti- 
tions. It  must  be  footed,  however,  how  often,  if  at  all,  the 
360°  point  is  passed.  Now,  if  the  telescope  is  plunged,  the 
plates  turned  180°  in  azimuth,  and  repetitions  of  the  angle 
again  be  made,  beginning  at  the  second  point,  the  mean  of  the 
two  sets  of  readings  will  give  still  more  nearly  the  true  angle, 
since  the  errors  of  adjustment  and  twist  of  station  are  thus 
lessened  and  those  of  observation  reduced. 

161.  By  Series.  Observe  as  before  upon  any  point,  and  read 
the  verniers,  clamp  the  lower  plate,  turn  the  vernier  plate  until 
the  telescope  may  be  fixed  upon  another  point,  and  again 
read ;  thus  continue  to  make  observations  upon  each  point 
desired  in  their  order,  sweeping  round  the  horizon,  and  make 
the  last  observation  upon  the  first  point.  The  last  reading 
should  be  the  same  as  the  first.  Plunge  the  telescope,  move 
the  plates  in  azimuth,  and  observe  on  the  points  again,  pro- 
ceeding in  the  contrary  direction.  Several  series  of  observa- 
tions may  thus  be  made,  as  in  the  method  by  repetition.  The 
magnitude  of  each  angle  is  obtained  from  the  mean  of  its 
reading. 

RKMAEiK.  Care  should  be  exercised  to  have  the  instrument 
properly  centred,  that  is,  set  precisely  over  the  centre  of  the 
station,  especially  if  the  object  sighted  is  near  the  observer. 
The  error  arising  from  an  eccentric  setting  is  inversely  as  the 
distance  of  the  object  sighted  ;  an  eccentric  setting  of  one  inch 
producing  an  error  of  nearly  three  (3')  minutes  of  arc  in  sight- 


108  PLANE   SURVEYING. 

ing  100  feet,  while  the  error  arising  from  a  sight  of  900  feet 
is  less  than  oue-tln'rd  (£')  of  a  minute. 

Read  both  verniers  to  eliminate  eccentricity.  See  that  the 
reading  is  not  made  from  the  wrong  end  of  the  vernier,  and  that 
a  half-degree  is  not  omitted,  calling  the  reading,  say,  36°  15', 
instead  of  36°  45'.  If  great  accuracy  is  required  when  running 
a  straight  or  broken  line,  lessen  errors  of  adjustment  by  re- 
versing the  instrument  in  altitude  and  azimuth,  making  two 
sets  of  observations  at  each  station,  and  take  the  mean  of  their 
readings.  See  Article  157. 

If  it  is  desired  to  locate  the  lines  surveyed  with  reference  to 
the  meridian,  the  bearing  of  one  of  them  should  be  taken  b}T 
the  needle  of  the  instrument ;  the  bearings  of  the  others  may 
be  deduced  therefrom.  See  Article  167. 

162.   Angle  of  Deflection.     The  amount  of  divergence  which 
a  line  makes  with  the  preceding  is  called  the  deflection,  and  the 
angle  which  measures  it  is  termed  the  deflection  angle. 
L~ 


In  the  figure  POM  is  the  deflection  angle  :  it  is  evidently  the 
supplement  of  LOP.  To  measure  it,  set  the  transit  at  0,  sight 
to  L,  clamp  the  limb  to  the  spindle  and  the  plates  together, 
then  plunge  the  telescope  :  it  will  point  to  M.  Take  the  reading, 
unclamp  the  vernier-plate  and  move  it  until  the  wires  intersect 
P.  The  difference  between  the  reading  now  and  the  first  read- 
ing is  the  deflection  angle.  If,  when  making  the  first  observa- 
tion, the  vernier  was  at  zero,  the  reading,  after  sighting  P, 
would  indicate  at  once  the  angle. 

163.    Traversing,  or  surveying  by  the  back  angle,  is  a  method 


ANGLES    WITH   TRANSIT. 


109 


of  surveying  by  which  the  direction  of  each  line  of  a  survey  is 
compared  with  the  first  as  a  meridian  or  reference  line.     It  is 

effected  as  follows : 

^*" 
P 


Let  it  be  required  to  traverse  the  broken  line  LMNOPQ. 
Setup  the  instrument  at  M,  clamp  the  vernier  at  zero,  for  con- 
venience, and,  with  the  lower  motion,  sight  L,  clamp  below, 
transit  the  telescope,  loosen  above  and  observe  N:  the  reading 
will  show  the  angle  M'MN  which  the  line  MN  forms  with  LM. 
Clamp  the  plates,  move  to  N,  plunge  the  telescope,  and,  with 
the  lower  motion,  sight  Jf,  the  index  remaining  as  at  M ;  then 
clamp  below,  loosen  above,  transit  the  telescope,  and  direct  it  to 
O :  the  index  will  show  the  angle  which  the  line  NO  makes  with 
LM.  And  so  continue  until  the  end  of  the  line. 

To  guard  against  mistakes  in  reading,  and  to  avoid  recording 
whether  the  deflection  is  right  or  left,  it  is  well  to  assume  all 
angles  measured  in  the  same  direction.'  In  the  figure  the 
readings  are  all  to  the  right,  or  clockwise,  as  indicated  by 
the  circular  arcs,  and  the  record  is  as  follows : 


STATIONS. 

AZIMUTHS 

WITH  LM. 

BEARINGS 

WITH  LM. 

MAGNETIC  BEARINGS 
ASSUMING  BEARING  OP 
LM  N.  50°  E. 

L 

0° 

North. 

N.  50°  E. 

M 

18° 

N.  18°  E. 

N.  68°  E. 

N 

340° 

N.  20°  W. 

N.  30°  E. 

0 

360°  or  0° 

North. 

N.  60°  E. 

P 

90° 

East. 

S.  4.0°  E. 

From  the  nature  of  the  operation  it  may  be  perceived  that, 
algebraically,  the  azimuth  of  any  line  is  equal  to  its  deflection 


HO  PLANE   SURVEYING. 

plus  the  azimuth  of  the  preceding  line.  This  method  is  partic- 
ularly adapted  to  surveying  roads,  streets,  water  courses,  etc., 
and  even  in  farm  surveying  it  possesses  an  advantage  over  the 
survey  by  interior  angles,  on  account  of  the  readiness  it  affords 
in  obtaining  the  bearings  from  the  azimuths,  and  the  greater 
rapidity  with  which  the  work  may  be  plotted,  since  the  angle 
which  each  line  makes  with  the  assumed  meridian,  or  reference 
line,  is  taken  at  once  from  the  field  notes. 

Suppose  LM  in  the  figure  to  be  the  meridian  of  the  survey, 
and  the  azimuths  of  the  several  lines  as  recorded  in  the  table. 
Now,  assuming  the  direction  of  LM  to  be  north,  it  is  evident 
that  MN  will  be  in  the  northeast  quadrant  18°  from  the  north 
point,  or  N.  18°  E ;  NO  will  be  20°  to  the  wost  of  north,  or  N. 
20°  W. ;  OP,  making  no  angle  with  the  meridian,  will  have  a 
bearing  north,  and  PQ  east. 

So  that,  in  general, 

When  the  azimuth  is  less  than  90°,  it  equals  the  bearing,  and 
the  line  is  in  the  northeast  quadrant. 

When  the  azimuth  is  between  90°  and  180°,  the  bearing  is 
southeast,  and  is  the  supplement  of  the  azimuth. 

When  the  azimuth  is  between  180°  and  270°,  the  bearing  is 
southwesterly,  and  may  be  found  by  subtracting  180°  from  the 
azimuth. 

When  the  azimuth  is  between  270°  and  360°,  the  bearing 
is  northwesterly,  and  is  the  difference  between  360°  and  the 
azimuth. 

When  the  azimuth  is    90°,  the  bearing  is  due  east. 

When  the  azimuth  is  180°,  the  bearing  is  due  south. 
'  When  the  azimuth  is  270°,  the  bearing  is  due  west. 

When  the  azimuth  is  360°,  the  bearing  is  due  north. 

If  it  is  required  to  find  the  magnetic  or  true  bearing  of  any  or 
all  the  lines,  take  the  magnetic  or  true  bearing  of  the  meridian  of 
the  survey  and  apply  it,  by  addition  or  subtraction,  according  as 
the  bearing  of  the  assumed  meridian,  or  standard  line,  is  north- 
east or  southwest.  In  the  example  given,  suppose  the  bearing 
of  the  assumed  meridian  LM  to  be  N.  50°  E. :  then  the  bearing 


ANGLES    WITH   TRANSIT. 


ill 


of  the  second  line  MN  will  be  recorded  18°  to  the  east  of  the 
reference  line,  or  N.  68°  E. ;  the  line  NO,  having  a  deflection  of 
20°  to  the  left  of  the  reference  line  will  be  recorded  N.  30°  E. ; 
and  OP,  N.  50°  E.  Thus  the  fourth  column  is  added  to  the  table. 

164.   To  Traverse  a  Road,  as  LMNO.     Proceed  as  indicated 
in  the  last  article,  and  in  addition  measure  the  lines  LM, 
NO,  and  perpendicular  offsets  thereto,  at  proper  distances. 


N     . 

If  the  road  deviates  much  from  a  straight  line,  it  will  be 
necessary,  in  order  to  obtain  more  correctly  the  area,  to  take 
two  offsets  at  M,  one  perpendicular  to  LM,  the  other  to  MN\ ' 
and  also  two  at  N,  one  perpendicular  to  MN,  and  the  other 
perpendicular  to  NO.* 

Likewise  to  Survey  a  Small  Stream.  Traverse  and  measure 
the  distances  between  assumed  stations,  as  L,  M,  N,  O,  P,  so 
chosen  as  to  make  no  more  of  them  than  is  consistent  with  few 
and  short  offsets  to  the  various  bends  of  the  stream.  If  the 


stream  is  small,  not  exceeding  10  feet  in  width,  or  even  wider  if 
shallow,  and  it  is  desired  to  survey  it  between  X  and  Y,  a  good 
plan  is  to  run  a  straight  line  between  these  points  and  measure 
offsets  therefrom  to  the  stream ;  or,  if  such  a  line  will  make  the 
offsets  rather  long,  run  RQ,  and  measure  offsets  from  it  to  X 
and  Y"and  intermediate  points.  If,  however,  the  stream  is  wide 

*  Article  234. 


112 


PLANE   SURVEYING. 


and  the  crossing  difficult,  it  will  probably  be  better  to  use  more 
stations,  as  shown  in  the  figure.  If  a  compass  is  used,  the 
bearings  may  be  taken  instead  of  the  angles. 

If  a  river  of  considerable  width  is  to  be  surveyed,  it  will  be 
necessary,  in  addition  to  the  measurement  of  broken  lines  on 
each  side  from  which  offsets  are  taken,  to  make  a  series  of  an- 
gular measurements  connecting  the  lines  on  one  side  with  those 
on  the  other,  and  thence  by  trigonometrical  calculations  deter- 
mine their  relative  positions,  and  ultimately  the  surface  of  the 
river. 


C.    PROBLEMS  ON  ANGLES  AND  BEARINGS. 

165.   Angles  between  Lines.     To  determine  the   angle  be- 
tween two  lines,  meeting  at  a  point,  given  by  their  bearings. 


•E     W- 


1.  If  the  lines  run  between  the  same  cardinal  points,  that  is, 
in  the  same  quadrant,  take  the  difference  of  their  bearings. 

Suppose  the  bearing  of  OP  is  N.  32°  W.  and  that  of  OQ 
N.  60°  W. ;  the  angle  between  them  is  obviously  NOQ  -  NOP-, 
or,  60°  -  32°  =  28°. 

2.  When  the  lines  run  in  different  quadrants  and  both  above 
or  both  below  the  horizontal  or  E.  and  W.  line,  take  the  sum 
of  tlreir  bearings.     If  OP  bears  N.  60°  E.  and  OL  N.  20°  W., 
the  angle  POL  =  PON+  NOL  =  60°  +  20°  =  80°. 


PROBLEMS   ON   ANGLES   AND   BEARINGS. 


113 


3.  If  the  lines  run  in  diagonally  opposite  quadrants,  subtract 
the  difference  of  the  bearings  from  180°.  Assuming  the  bear- 
ing of  OP  N.  28°  E.  and  of  OL  S.  58°  W.,  the  angle 

POL  =  180°  -  LOM  =  180°  -  (58°  -  28°)  =  150°. 


4.  When  the  lines  are  in  different  quadrants,  and  both  to  the 
right  or  both  to  the  left  of  the  vertical  or  N.  and  S.  line,  sub- 
tract the  sum  of  the  bearings  from  180°.  If  OP  bears  N.  65° 
E.  and  OL  S.  42°  E.,  the  angle 

POL  =  180°  -  (NOP  +  SOL)  =  180°  -  (65°  +  42°)  =  73°. 

ADDITIONAL  EXAMPLES. 

1.  A  line  OP  bears  N.  40°  W.  and  OL  N.  40°  E.,;   required 
the  angle  POL. 

2.  Find  the  angle  POL,  when  OP  bears  S.  50°  E.  and  OL 

N.  89°  E. 

3.  Required  the  angle  at  0,  when  OP  bears  N.  80°  W.  and 
OL  S.  79°  E. 

4.  What  is  the  angle  O,  if  OP  runs  S.  89|°  W.  and  OL 
N.  89£°  E.  ? 

5.  A  line  OP  runs  S.  70°  W.  and  OL  S.  45°  W.      Find  the 
angle  0. 


114 


PLANE    SURVEYING. 


166.  There  may  be  given  the  bearing  of  a  line,  as  MO,  and 
the  deflection  angle  LOP,  to  the  right  or  left  of  the  direction 
of  MO,  to  find  the  bearing  of  OP ;  or,  the  bearings  of  MO  and 
OP  may  be  given  to  determine  the  magnitude  of  the  deflection 
angle  LOP. 


a.  Given  the  bearing  of  a  line  and  the  deflection  of  the  next, 
to  find  its  bearing. 

Suppose  MO  bears  N.  32°  W.,  and  the  deflection  of  OP=  20° 
to  the  left ;  the  bearing  of  OP  is  evidently  20°  farther  towards 
the  west  than  MO  or  its  prolongation  OL.  It  is  therefore 
N.  52°  W.  Again,  assuming  RO  bears  N.  60°  E.  and  the  de- 
flection of  OQ  40°  to  the  right,  it  is  evident  that  OQ  is  in  the 
southeast  quadrant,  10°  from  the  east  point ;  or,  its  bearing 
is  S.  80°  E. 

6.  When  the  bearings  of  the  lines  are  given,  to  determine  the 
deflection. 

Suppose  LO  (p.  115)  bears  N.  20°  E.  and  OM  N.  70°  E. ;  the 
deflection  of  03/from  LO,  or  its  prolongation  OP,  is  evidently 
70°  _  20°  =  50°  to  the  right.  Again,  the  bearing  of  LO  re- 
maining the  same,  .and  that  of  OQ  N.  30°  W.,  then  it  is  readily 
seen  that  the  deflection  angle  is  20°  +  30°  =  50°  to  the  left. 


PROBLEMS  ON  ANGLES  AND  BEARINGS. 

JT 

,P 


115 


General  rules'  might  be  given  for  the  cases  under  the  above 
heads,  corresponding  to  those  in  the  preceding  article,  but  they 
are  deemed  unnecessary,  as  a  little  reflection  will  enable  the 
student  to  determine  the  required  bearing,  or  angle,  in  any 
given  case. 

167.  Given  the  angle  between  two  lines,  and  the  bearing  of 
one  line,  to  find  the  bearing  of  the  other. 

The  solution  of  this  problem  is  ordinarily  required  in  transit 
surveying,  for,  when  surveying  with  that  instrument,  it  is  com- 
mon to  take  the  bearing  of  only  one  line,  and  deduce  the 
courses  of  the  others  from  that  bearing  and  the  measured 
angles.  Suppose  LO  bears  N.  24° 
W.  and  the  angle  LOP=  82°,  to 
find  the  bearing  of  OP.  It  is  evi- 
dent that  the  bearing  of  OP  or  the 
angle  NOP,  which  gives  the  degrees 
in  the  bearing, 


=  180°- (24° +  82°)  =  74°. 
Hence  the  bearing  of  OP  is  N.  74°  E. 

Assume  the  angle  POM=  100°, 
and  the  bearing  of  OP  as  found 
above;  then,  since  there  are  100° 
—  74°,  or  26°,  more  in  the  angle  than  lies  between  OP  and  the 


116  PLANE   SURVEYING. 

north  point,  the  position  of  OM  is  to  the  west  of  north  26°,  or 
its  bearing  is  N.  26°  W. 

Some  simple  combinations,  as  indicated  in  the  illustrations 
given,  will  enable  the  student,  unencumbered  with  rules,  to 
readily  solve  any  of  the  problems  coming  under  this  head. 

EXAMPLES. 

1.  A  line  bears  S.  89°  15'  W.     What  is  the  bearing  of  a  line 
perpendicular  to  it?    Also,  the  bearing  of  a  line  making  an  angle 
of  135°  with  it?    Is  there  more  than  one  answer  to  the  last? 

2.  If  OP  bears  S.  36°  W.,  and  the  angle  0^=68°,  what 
is  the  bearing  of  PL?  Ans.   N.  32°  W. 

M  SUGGESTION.      Pass     a     meridian 

?O  through  the  angle,  and  consider  the 
given  bearing  reversed. 

3.  The  angles  L,  M,  0,  P,  of  the 
trapezium  are  respectively  62°,  130°, 
80°,  and  88°,  and  the  bearing  of  LM 
N.  70°  E. ;  find  the  other  bearings.* 

168.  To  Change  the  Bearings  of  the  Sides  of  a  Survey.  It  is 
sometimes  desirable  to  change  the  bearings  of  a  survey  so  that 
a  particular  side  shall  become  a  meridian.  The  whole  plat  is 
conceived  to  revolve  through  an  angle  sufficient  to  make  the 
desired  side  the  meridian ;  the  relative  position  of  the  sides 
remains  unaltered.  The  following  rule  is  substantially  that 
given  by  Gummere,  who  states  that  the  method  was  communi- 
cated to  him  by  Prof.  Robert  Patterson,  late  of  Philadelphia. 

RULE. 

Subtract  the  bearing  of  the  side  that  is  to  be  made  a  meridian 
from  those  bearings  that  are  between  the  same  points  that  it  is, 

*  The  calculation  may  be  tested,  after  having  deduced  the  bearings  of 
all  the  sides,  by  taking  the  last  bearing  found,  as  PL,  applying  the  angle 
L,  and  observing  if  it  gives  the  proper  bearing  of  LM. 


PROBLEMS    ON    ANGLES  AND   BEARINGS. 


117 


and  also  from  those  that  are  between  points  directly  opposite  to 
them.  If  it  is  greater  than  any  of  those  bearings,  take  the  differ- 
ence, and  change  west  to  east  or  east  to  west. 

Add  the  bearing  of  the  side  tvhich  is  to  be  made  a  meridian  to 
those  bearings  tvhich  are  neither  between  the  same  points  that  it  is 
nor  between  the  points  directly  opposite  to  them.  If  either  of  the 
sums  exceed  90°,  take  the  supplement,  and  change  south  to  north 
or  north  to  south,. 

The  accompanying  diagram  of  full  and  dotted  lines  exhibits 
the  positions  of  the  sides  of  the  following  described  farm,  be- 
fore and  after  turning  through  16^°  to  the  right: 

(1)  N.  16^°  W.,  24.63  chains  ;     (3)    S.  ±°  W.,  34.28  chains ; 

(2)  S.  79°  W.,  27.00  chains  ;        (4)  .N.  65°  E.,  37.20  chains, 
to  the  place  of  beginning.     The  bearings  are  changed  so  as  to 
make  the  first  side  a  meridian. 


EXAMPLES. 

1.    Given  the  bearings  of  a  tract  of  land : 
(1)    S.  10°  E. ;          (2)    S.  30°  W. ;         (3)    N.  60°  W. ; 
(4)    N.  20°  W. ;         (5)    N.  80°  E., 


118  PLANE   SURVEYING. 

to  the  place  of  beginning.     Required  the  changed  bearings  that 
the  fourth  side  may  be  a  meridian. 

(1)  S.  10°  E.  (4)    North. 

20 

Changed  bearing,  S.  10°  W. 

(2)  S.  30°  W.  (5)    N.  80°  E. 

20  20 


Changed  bearing,  S.  50°  W.  100 

(3)    N.  60°  W. 


20  Changed  bearing,    S.  80°  E. 

Changed  bearing,   N.  40°  W. 

The  student  who  avails  himself  of  the  hints  and  methods 
referring  to  the  manipulation  of  angles  and  bearings  as  given 
in  the  preceding  articles,  will  have  no  difficulty  in  determining 
the  changed  bearings  direct  from  the  data,  without  the  use  of 
rules.  Thus  in  the  example  above  it  will  be  observed  that  each 
line  is  turned  through  20°  to  the  right;  that  is,  the  fourth 
course  is  made  due  north.  The  next  side  to  it  going  round  to 
the  right,  N.  80°  E.,  will  be  turned  the  same  number  of  de- 
grees (20),  which  places  it  10°  from  the  east  point  in  the  south- 
east quarter,  or  its  bearing  is  S.  807  E.  ;  the  first  side  turning 
through  the  same  angle  (20°)  will  be  thrown  10°  west  of  the 
south  point,  or  S.  10°  W.  ;  the  second  course  will  be  20°  farther 
to  the  southwest,  or  S.  50°  W. ;  and  the  third  course  turned 
toward  the  north  point  20°  will  be  N.  40°  W. 

2.  Find  the  bearings  of  all  the  sides  of  the  following  de- 
scribed tract  of  land  when  the  second  side  is  made  a  meridian  : 

(1)  N.  681°  E.,  8.42  chains  ;        (3)    S.  78f°  W.,  4.90  chains; 

(2)  N.  27°  W.,  10.25  chains  ;       (4)    S.  1°  E.,  4.40  chains  ; 

(5)    S.  12°  E.,  7.04  chains, 
to  the  place  of  beginning. 


PROBLEMS  ON  ANGLES  AND  BEARINGS.      119 

3.  Given  the  bearings  of  a  tract  of  land  as  follows : 

(1)  S.  39i°  W. ;       (3)    N.  15°  W. ;         (5)    N.  2°  E. ; 

(2)  East ;  (4)    N.  79£°  E. ;        (6)    S.  73f°  W., 

to  find  the  bearings  of  all  the  sides  when  the  first  becomes  a 
meridian. 

4.  Given  the  bearings  of  a  tract  of  land  as  follows : 

(1)  S.  79°  W.  ;         (3)    N.  89L°E. ;         (5)    S.  80f  E. ; 

(2)  S.  £°W.  ;  (4)    N.  IfE.  ;  (6)    S.  581°  E. ; 

(7)    N.  39°  E. ;         (8)    N.  16}-°  W., 
to  find  the  bearings  when  the  eighth  side  becomes  a  meridian. 

EXERCISES. 

1.  With  a  transit,  using  back  and  fore  sights,  run  a  tangent 
forward  and  back  over  hilly  and  brush  land  requiring  six  or 
eight  settings  of  the  instrument.     The  last  two  points  set  for- 
ward will  give  the  direction  back.     Note  the  distance,  if  any, 
between  the  corresponding  positions  occupied  by  the  instru- 
ment. 

2.  Traverse,  or  survey  by  the  back  angle,  a  broken  line  of 
six  stations,  using  the  first  line  as  the  meridian,  or  reference 
line,  of  the  survey.     Record  the  notes,  indicating  the  azimuthal 
angles  and  bearings. 

3.  Measure  the  three  angles  of  a  triangular  piece  of  land, 
the  cornei-s  being  visible  from  each  other ;  see  how  much,  if 
any,  their  sum  differs  from  two  right  angles. 

4.  Traverse  a  pentagonal  field,  the  index  at  the  beginning 
being  sot  nt  zero,  and  see  if,  when  finally  sighting  on  the  station 
first  occupied,  the  reading  is  zero. 


120  PLANE   SURVEYING. 

SECTION   III. 

OBSTACLES. 
A.   PROBLEMS  ON  PERPENDICULARS  AND  PARALLELS. 

169.  The  Obstacles  which  occur  in  field  work  are  more  easily 
and  expedittously  overcome  with  the  compass,  or  transit,  and 
chain,  than  with  the  chain  alone.     Methods  for  the  latter  were 
given  and  illustrated  in  Chapter  I.  Section  II.,  Chain  Survey- 
ing. 

To  erect  a  perpendicular  to  a  line  at  any  given  point.  Set  up 
the  instrument  over  the  point ;  if  a  compass  is  used,  take  the 
bearing  of  the  line,  and  then  move  the  instrument  in  azimuth 
until  a  bearing  differing  90°  from  the  first  is  observed.  The 
line  of  sights  will  then  indicate  the  direction  of  the  required 
perpendicular.  If  a  transit  is  employed,  centre  on  the  point, 
sight  to  a  point  in  the  line,  clamp  to  spindle,  and  turn  the  ver- 
nier plate  90°  either  way ;  then  the  line  of  collimation  will  show 
the  direction  of  the  perpendicular  sought.  Of  course  b}-  the 
methods  explained  above,  a  line  can  be  run  with  either  instru- 
ment from  any  given  point  and  making  any  given  angle  thereat 
with  a  line. 

170.  To  let  fall  a  perpendicular  from  a  given  point  to  a  line. 
Let  P  be  the  point,  and  LN  the  line.     If  the  compass  is  used, 

take  the  bearing  of  LN,  remove  the 
instrument  to  P,  and  with  a  bearing 
differing  90°  from  the  first,  run  PO 
for  the  required  perpendicular.  With 
— N  a  transit  centre  on  L,  measure  the 
angle  OLP,  remove  to  P,  and  make 

the  angle  LPO  equal  to  the  complement  of  L  ;  the  line  of  sight 
of  the  instrument  will  then  be  in  the  direction  of  the  required 
perpendicular. 


PERPENDICULARS  AND  PARALLELS. 


121 


171.  To  let  fall  a  perpendicular  to  a  line  from  an  inaccessible 
point.  Measure  the  distance  between  any  two  points,  as  L  and 
N,  in  the  line ;  also  the  angles 
PLN  and  LNP.  Then  in  the 
triangle  PLN  we  have  given  the 
side  LN  and  the  angles  to  find 
PL  or  PN.  Computing  PL, 
the  distance 

LO=PLcosPLO. 
Or  we  may  deduce  an  expression  for  LO  in  terms  of  the  meas- 
ured line  and  the  observed  angles,  thus  : 

LO=POcotPLO. 

NO  =  POcotPNO. 

LO:NO  =  cot  PLO :  cot  PNO, 

LO:LO  +  NO=  cot  PLO :  cot  PLO  +  cot  PNO ; 


jy 


LN  cot  PLO 


Hence 
and 

but 

therefore  

cot  PLO  +  cot  PNO 

QUERY.  Could  a  line  be  run  not  perpendicular  as  above 
through  an  inaccessible  point,  making  any  angle  with  the  given 
line? 

172.  To  run  a  line  through  a  given  point  parallel  to  a  given 
line.  With  the  compass  obtain  the  bearing  of  the  line,  and 
then  from  the  given  point  run  a 

line  with  the  same  bearing.     With     -<c N 

the  transit,  LN  being  the  line  and 

P  the  point,  centre  on  L,  measure 

the  angle  NLP,  remove  to  P,  and          R 

make  the  angle  LPR  equal  to  NLP;  the  line  of  collimation 

will  then  be  in  the  required  parallel. 


B.     PROBLEMS   ON   ALIGNMENT. 

173.    To  prolong  a  line,  as  LN,  beyond  a  tree,  a  building,  or 
any  obstacle. 


122 


PLANE   SURVEYING. 


First  Method.  By  Deflection  Angles.  Set  up  the  instrument 
at  any  point  of  the  line,  as  N,  and  deflect,  sufficient  to  pass  the 
obstacle,  to  any  point  P.  Measure  NP,  remove  to  P,  deflect 
to  0,  making  the  angle  QPO  double  the  angle  at  N. 


Measure  PO  =  PN,  place  the  instrument  at  0,  observe  P, 
plunge  the  telescope  and  deflect  to  R,  so  that  SOR  =  \  OPQ ; 
the  telescope  will  then  be  in  the  prolongation  of  LN. 

174.   Second  Method.     By  Equilateral  Triangle.     Deflect  60° 
from  the  direction  of  the  line  at  N;  measure  to  P  a  distance 


sufficient  that  PO,  making  an  angle  of  60°  with  P^,  will  clear 
the  obstacle.  Measure  PO  =  PN,  and  turn  the  telescope  in  the 
direction  of  OR,  the  prolongation  of  LN,  by  deflecting  60°  from 
the  direction  of  PO. 


175.    Third  Method.     By  Isosceles  Triangle.     Deflect  at  N 
45°  to  M,  measure  NM,  make  NMO  a  right  angle,  and  MO 

H 


=  MN;  at  0  turn  into  OR  by  deflecting  from  the  direction  of 
OM  45°. 


PROBLEMS   ON    ALIGNMENT. 


123 


176.   Fourth  Method.     By  Perpendiculars.     Erect  a  perpen- 
dicular NK  of  sufficient  length  that  a  line   passing  through 


K  parallel  to  LN  will  clear  the  obstacle ;  run  KM ;  lay  off 
MO  =  NK,  and  a  right  angle  turned  from  MO  will  indicate  the 
direction  of  LN,  or  its  prolongation  OR. 

177.   Random  Line.     When  brush,  wood,  or  any  obstruction 
prevents  N  being  seen  from  L,  run  a  line  LP  as  nearly  as  may 


be  judged  in  the  direction  of  LN:  when  opposite  N,  as  at  P, 
measure  the  shortest  distance  from  P  to  N,  call  it  d ;  then  the 


57.3  xd 
LP 


angle  PLN  in  degrees  = 

Setting  up  again  at  L,  and  applying  the  correction  thus 
found  in  a  proper  manner  to  the  angle  or  bearing  before  used, 
the  line  LN  may  be  traced. 


N 


Demonstration.  When  the  distance  PN  does  not  exceed  5 
per  cent  of  the  length  of  PL,  LN  and  PL  may  be  regarded  as 
radii  of  a  circle,  and  PN  coincident  with  the  arc  which  subtends 
the  angle  PLN',  then 


124  PLANE   SURVEYING. 

27r£P:360  =  PN:  PLN, 

360  x  PN     57.3  x  PN 


PLN. 


LP 


When  PN  exceeds  the  limit  stated,  the  angle  PLN  should  be 
found  by  measuring  PN  perpendicularly  from  PL,  and  dividing 
this  by  the  length  LP  for  the  tangent  of  the  angle  PLN. 

EXAMPLES. 

1.  A  random  line  was  run  N.  41°  15'  E.  18.34  chains,  when 
the  nearest  distance  to  the  desired  corner,  which  was  to  the  left, 
was  found  to  be  16  links.    Required  the  correction  and  the  bear- 
ing of  the  true  line.    Ans.  Cor.  30' ;  bearing  of  line,  N.  40°  45'  E. 

2.  A  random  line  was  run  S.  89°  45'  W.  24.80  chains,  when 
the  corner  was  found  22  links  to  the  right.    Find  the  correction 
and  the  bearing  of  the  line. 

3.  The  length  of  a  random  line  is  16.64  chains,  and  a  per- 
pendicular from  its  extremity  to  the  desired  point  equals  96 
links.     What  correction  is  needed  ? 

-P 

N 


M 


4.  A  random  line  LP,  25.12  chains  long,  run  by  transit, 
makes  an  angle  of  27°  with  LM,  and  the  point  P  is  18  links  to 
the  left  of  N',  LN  being  the  true  line.  Determine  the  proper 
angle  to  turn  off  at  L  with  which  to  trace  LN. 

C.     PROBLEMS   ON   MEASUREMENT. 

178.  a.  When  the  Ends  of  the  Line  are  Accessible  and 
Visible  from  Each  Other. 


PROBLEMS    ON    MEASUREMENT. 


125 


The  methods  indicated  in  Problems  on  Alignment  will  be 
found  useful  in  many  instances  for  the  determination  of  the 
lengths  of  lines,  the  direct  measurements  of  which  are  imprac- 
ticable. Thus,  in  the  figure  in  Article  176,  the  distance  NO 
will  be  found  by  measuring  KM. 

In  figure  accompanying  Article  174  the  measurement  of  either 
NP  or  PO  will  give  the  side  NO. 

Otherwise  (Article  175).  Measure  NM,  and  multiply  it  by 
V^2,  or  extract  the  square  root  of  twice  the  square  of  NM  for 
the  required  length  NO. 

By  random  line,  as  in  Article  177,  when  the  shortest  distance 
PN\s  taken,  the'length  of  the  true  line  will  equal  the  measured 
or  random  line. 

If  the  perpendicular  from  P  is  used,  then  the  length  of  the 
true  line  will  equal  the  square  root  of  the  sum  of  the  squares  of 
LP  and  PN-,  that  is, 

To  ascertain  the  horizontal  measurement  of  a  hillside,  take 
the  angle  of  its  slope,  measure  up  or  down  it  (preferably  down), 
and  the  product  of  this  distance  and  the  cosine  of  the  angle  will 
be  the  horizontal  distance  required.* 

179.   By  Triangulation.     Measure  LP  and  the  angles  L  and 
P ;  the  sine  proportion  may  then  be  em- 
ployed to  determine 


180.  Otherwise.  Measure  LP,  PN, 
and  the  angle  P.  Then  having  two  sides 
and  the  included  angle  of  the'  triangle, 
the  third  side  LN  may  be  computed. 


*  When  measuring  the  angle  of  elevation,  the  surveyor  should  sight  to 
a  point  on  the  rod  a  distance  above  ground  equal  to  the  height  of  the  line 
of  collimation  of  his  instrument. 


126  PLANE   SURVEYING. 

181.  b.  When  One  End  of  the  Line  is  Inaccessible.     Let  N 

represent  the  inaccessible  but 
visible  end  of  the  line  LN,  the 
length  of  which  is  desired.  Meas- 
ure LP  of  such  length,  if  possible, 
that  none  of  the  angles  will  be  less 
than  30°  ;  the  nearer  LNP  is  equi- 

lateral, the  better.    Observe  the  angles  L  and  P.    Then,  by  the 

sine  proportion, 


sin  (L  +  P) 

182.  When  the  Points  are  not  Visible  from  Each  Other.    In 

the  figure  let  N  represent  the  invisible  point  in  the  line  LN, 
the  length  of  which  is  required. 
Measure  a  line  in  any  convenient 
direction  through  L,  as  MP,  not- 
'ing  the  distances  ML  and  LP,  of 
such  a  length  that  the  point  .2V 
may  be  seen  from  each  extrem- 
ity. Observe  the  angles  P  and 

M.    In  the  triangle  PMN,  find, 
f  ii» 

-kj  by  the  sine  proportion,  the  length 

of  PN.  Then  in  PNL  are  known 

•  two  sides  and  the  included  angle, 

0  with  which  may  be  found  LN. 

It  will  be  observed  that  the  problem  requires  the  measure- 

ment of  the  distance  between  two  points,  L  and  N,  invisible 

from  each  other,  and  direction  unknown.     If  it  were  simply  to 

determine  the  distance  from  L  to  an  invisible  point  in  the  pro- 

longation of  OL,  we  should  measure  perpendicularly  from   OL 

to  a  point  P,  from  which  the  point  N  could  be  seen,  observe 

the  angle  LPN;  then  LN=  PL  x  tan  LPN. 

QUERY.  What  would  be  the  best  method  of  solving  the  prob- 
lem under  the  last  supposition,  if  it  were  impracticable  to 
measure  a  perpendicular  from  OL? 


PROBLEMS   ON    MEASUREMENT. 


127 


Let  it 


183.  c.  When  the  Ends  of  the  Line  are  Inaccessible, 
be  required  to  determine  the  length 

of  the  inaccessible  line  LN.  Meas- 
ure OP,  and  observe  the  angles  LON, 
NOP,  OPL,  and  LPN;  then  in  the 
triangle  LOP  compute  LO,  and  in 
NOP,  ON.  There  will  then  be  given 
two  sides  and  the  included  angle  of 
the  triangle  LON  to  find  LN. 

184.  The  same  general  method  would  apply  if  the  base  inter- 
sected the  line  the  length  of  which  is  desired.     Suppose  it  is 
required  to  determine  the  distance  be- 
tween L  and   N,  points  on   opposite 

sides  of  two  inlets,  J^and  T.  Meas- 
ure OP  and  take  the  angles  at  the  ex- 
tremities on  both  sides  of  the  base. 
There  will  then  be  data  sufficient  to 
find  OL  and  ON,  and  finally  LN. 


QUERY.  Would  it  be  practicable  in 
any  case  to  make  OP  perpendicular  to 
LN?  If  so,  would  it  be  necessary  to 
measure  the  distance  OP  and  all  the 
angles,  as  above?  Why? 


N 


EXAMPLES. 

1.  To  determine  the  distance  between  two  points  L  and  2V, 
on  opposite  banks  of  a  stream,  I  measured  a  base  LP  =  300 
feet,  and  observed  the  angles  which  N  made  with  L  and  P  to 
be  58°  45'  and  64°  50',  respectively.     Required  LN. 

2.  If  LP  in  Example  1  were  taken  at  right  angles  to  LN, 
the  angle  P  being  40°  30',  what  would  be  the  length  of  LN? 

3.  To  ascertain  the  distance  LN  between  two  inaccessible 
points  invisible  from  each  other,  I  measured  a  line  MP  through 


128  PLANE   SURVEYING. 

L,  from  the  extremities  of  which  ^7"  could  be  seen.  ML  =  160 
feet ;  LP=  200  feet ;  angle  at  M  =  65°  30' ;  angle  P=  69°  15'. 
What  is  the  length  of  LN? 

4.  To  determine  the  distance  between  two  points  L  and  N, 
situated  on  the  side  of  a  river  opposite  to  where  I  was,  a  base 
line  OP  400  feet  long  was  measured,  and  the  following  angles 
observed  :  LON=  68°  30' ;  NOP  =  32°  45' ;  NPL  =  50°  30' ; 
LPO  =  40°  15'.  Required  LN. 


EXERCISES. 

1.  Prolong  a  line  beyond  a  house,  tree,  or  other  obstruction, 
using  any  one  of  the  methods  herein  given.     Return,  pass  the 
obstruction  by  some  other  method.     See  how  near  the  starting- 
point  is  reached. 

2.  Run  a  trial  line  of  considerable  length  through  a  wood, 
with  a  view  of   sighting  a  stake  previously  set.      Make   the 
proper  measurements  and  calculation  to  correct  the  angle  and 
re-run  the  line.     Note  the  distance,  if  any,  from  the  stake  after 
the  second  trial. 

3.  Triangulate  across  a  creek  or  small  lake.     Use  at  least 
two  methods.     See  how  near  the  results  agree. 

4.  By   triangulation    determine   the   distance   between    two 
points  without  going  near  them.     Verify  the  result  by  subse- 
quent measurement. 

5.  Measure  the  distance  between  two  points  in  a  given  line, 
invisible  and  assumed  inaccessible  from  each  other.     Compare 
the   results  of   two  methods.     Verify  subsequently  by  direct 
measurement. 

6.  Run  a  trial  line  between  two  points  which  are  invisible 
from  each  other,  on  account  of  an  intervening  ridge.     Correct 
the  angle  and  re-run  the  line.    If  the  proper  point  is  not  reached, 
should  the  angle  be  again  corrected? 


ACCESSIBLE   HEIGHTS.  129 

SECTION  IV. 
HEIGHTS  AND  DISTANCES. 

A.     ACCESSIBLE   HEIGHTS. 

185.  Let  it  be  required  to  determine  the  height  P  above  a 
horizontal  plane  LN.  Measure  the  distance  LN  and  the  angle 
of  elevation  L.  Then 


If  the  ground  is  level,  or  nearly  so,  the 
telescope   cannot  be   placed   at   L,  in  the  / 

horizontal  plane  with  JV,  but  at  some  point  /'/ 

I,  and  the  angle  Pin  is  measured  instead  of        /'/' 
PLN.     In  such  a  case  Nn  must  be  added     l[/~ 
to  the  calculated  height. 


186.  Let  it  be  required  to  find  the  height  of  an  object  stand- 
ing on  an  inclined  plane  ON.  Measure 
the  distances  NL  and  LO,  and  the  angles 
NLP  and  NOP.  In  the  triangle  OLP, 
by  the  sine  proportion,  find  PL.  Then 
in  the  triangle  PLN,  having  two  sides 
and  the  included  angle,  PN  may  be 
determined. 


187.    Otherwise.   Measure  NL,  and  at 

L  the  angles  of  elevation  of  N  and  P.     Then  the  projection  of 
LNon  the  horizontal  plane 

=  LM=LNcosNLM, 
and         MN=  LN  sin  NLM ; 

PM=LMt&nPLM; 
whence    PN=  PM—  NM;  or,  expressed  in  a  single  equation, 

PN=  LN  X  cos  NLM  x  tan  PLM-LN  X  sin  NLM. 


130  PLANE  SURVEYING. 

EXAMPLES. 

1.  At  120  feet  distance  from  the  centre  of  the  foot  of  a 
liberty  pole,   the  angle  of   elevation   of  its   top  was  38°  40'. 
Required  its  height. 

2.  The  distance  LN  (see  Article  185)  measures  90  feet,  the 
angle  of  elevation  I  is  42°  30',  the  telescope  being  4.8'  above 
the  horizontal  plane  LN.    Determine  height  of  the  point  P. 

3.  To  determine  the  height  of  an  object  on  an  inclined  plane, 
two  stations,  L  and  0  (marginal  figure,  Article  186),  were  se- 
lected, one  50  feet  and  the  other  110  feet,  measured  on  the  slope 
from  N.     The  angle  NLP=40°  15',  and  NOP—  22°  30'.     Re- 
quired the  height. 

QUERIES.  Practically,  is  it  necessary  to  know  the  height  of 
instrument  *  in  such  cases  ? 

If  there  was  a  change  of  slope  at  L,  would  any  other 
measurement  be  necessary  to  calculate  the  required  height? 

4.  Suppose  NL  (figure,  Article  186)  measures  60  feet,  and 
the  angles  of  elevation   at  JO,  of  N  and  P,  are  respectively 
12°  30'  and  59°  20'.     Determine  the  height  of  P  above  N. 

B.    INACCESSIBLE  HEIGHTS. 

188.  To  determine  the  height  of  an  object  situated  on  an 
inaccessible  hill. 

Measure  in  the  same  vertical 
plane  with  P  a  horizontal  line 
LN,  and  observe  at  N  the  angles 
of  elevation  of  the  points  0  and  P, 
and  at  L  the  angle  of  elevation  of 
P.  In  the  triangle  LNP,  by  the 
sine  proportion,  calculate  PN. 

By  the  same  method,  find  NO 
from  the  triangle  PON.  Then 

*  Height  of  instrument  is  the  height  of  the  line  of  sight  above  the 
the  ground,  or  any  other  assumed  horizontal  plane 


INACCESSIBLE   HEIGHTS.  131 

PO  =  PN  sin  PNM -  NO  sin  ONM. 

The  student  may  show,  after  finding  PN  and  NO  as  above, 
a  different  method  of  finding  PO  than  that  indicated. 

Ex.  At  a  certain  station  the  angle  of  elevation  of  the  base 
of  a  tower  on  a  hill-top  was  38°  40',  and  that  of  the  top 
50°  15' ;  190  feet  more  remote,  the  angle  to  the  top  was  36°  20'. 
The  stations  being  in  the  same  horizontal  plane,  required  the 
height  of  tower  and  of  the  hill. 

189.  Let  PO  be  an  object  whose  height  is  required.     Measure 
in   the   same   vertical   plane   with   P  a 

horizontal  base  line  LN,  and  observe 
the  angles  of  elevation  PLN  and  PNO. 
Then,  by  the  sine  proportion,  find  PN, 
and 

PO  =  PN8\nPNO. 

190.  Otherwise.   PO  cot  L  =  LO, 


LO-NO  =  LN\ 
or,  PO  (cot  L  -  cot  N)  =  LN. 

,,PO==          LN 

cot  L  —  cot  N 

Ex.   If  LN=  120  feet,  and  the  angles  at  L  and  N  respec- 
tively 27°  50'  and  45°  19', 

PO  = — =136.6  feet.     Ans. 

cot 2 7°  50' -cot 45°  19' 

191.  If  it  is  impracticable  to  locate  the  base  line  in  a  hori- 
zontal plane,  measure  from  L  in  the 
direction  of  P  any  line  LN,  and  at  L 
take  the  angles  of  elevation  of  N  and  P. 
Observe  also  the  angle  at  N.  By  the 
sine  proportion  obtain  LP.  Then 

PO  =  LPs\nPLO, 
and          PR  =PO-RO  =  LP  sin  PLO  -  Z^sin  NLO. 


132  PLANE   SURVEYING. 

QUERY.     May  the  observed  angle  at  N  be  either  LNP  or 
PNR? 

192.   Otherwise.     L  and  N  being  in  different  planes,  measure 
the  horizontal  distance  between  them.     Observe  the  angle  of 
elevation  PLO  and  the  horizontal 
P  angles  OLN  and  ONL.      By  the 

sine  proportion  find  LO.    Then 


or,  expressed  in  a  single  equation, 

po  =  LNsm  LNO  tan  PLO 

sin  NOL 

which  equals  the  height  of  P  above 
the  horizontal  plane  through  L. 

If  it  is  required  to  find  the  height  of  P  above  the  horizontal 
plane  through  JV,  proceed  as  follows  :  Assuming  N  to  be  be- 
low *  L,  observe  at  N  the  angle  of  elevation  of  P ;  then  find  the 
horizontal  distance  between  N  and  0  by  the  sine  proportion, 
using  the  triangle  NLO ;  thus,  sin  0 :  sin  L  =  LN:  fourth  term. 
This  fourth  term  will  not  be  NO.  since  the  measurement  of  the 
distance  and  angles  employed  in  the  computation  is  referred 
to  a  horizontal  plane,  and  hence  the  fourth  term  will  express  the 
horizontal  distance  between  N  and  0,  which  equals  NR,  R  being 
a  point  in  the  prolongation  of  the  vertical  PO.  Whence, 
PR  =  NRtenPNR. 

EXAMPLES. 

1.  At  a  certain  station  the  angle  of  elevation  of  the  top  of 
an  inaccessible  object  situated  on  a  horizontal  plane  was  60°  50', 
and  120  feet  farther  away  the  angle  was  29°  10'.     Required  the 
height  of  Jthe  object  and  its  distance  from  the  first  station. 

2.  Suppose  LN  (figure,  Article  191)  is  140  feet,  the  angles  of 
elevation  at  L,  of  N  and  P,  are  respectively  9°  25'  and  30°  16', 

*It  may  obviously  be  above  or  below  L ;   the  same  reasoning  will  hold 


INACCESSIBLE   HEIGHTS. 


and  the  angle  PNR  =  42°.     Find  the  height  of  P  above  0 
and  R. 

3.   In  figure,  Article  192,  suppose 

LN=  1000  feet ;  angle  PLO  =  26°  18' ; 

angle  OLN=  36°  20' ;  angle  PNR  =  55°  10'. 

angle  ONL  =  95°  40' ; 
Find  PO  and  RO. 

193.  To  determine  the  perpendicular  distance  from  a  given 
horizontal  plane  of  an  inaccessible  object  situated  below  it. 

Let  P  be  the  point  whose  perpendicular  distance  from  a  hori- 
zontal plane  through  L  is  required.  Select 
two  points  L  and  N  visible  from  each  other, 
and  from  which  P  can  be  seen.  Measure 
the  horizontal  distance  between  them ;  ob- 
serve also  the  horizontal  angles  PLN  and 
PNL,  and  the  angle  of  depression  of  the 
point  P,  at  L.  By  the  sine  proportion  cal- 
culate the  horizontal  distance  from  L  to  P; 
this  multiplied  by  the  tangent  of  the  angle 
of  depression  observed  at  L  will  give  the  perpendicular  dis- 
tance required. 

If  L  and  N  are  not  in  the  same  horizontal  plane,  observe  at 
N  the  angle  of  depression  of  P,  and  calculate  as  above  the  per- 
pendicular distance  between  the  point  and  the  horizontal  plane 
through  N.  The  difference  of  these  perpendicular  distances 
will  also  give  the  difference  in  height  of  L  and  N.  A  check  on 
the  work  may  be  had  by  determining  from  more  direct  methods 
already  given  the  difference  in  elevation  of  L  and  N. 

EXAMPLES. 

1.  At  L  and  N  (last  figure)  the  horizontal  angles  measure 
respectively  67°  40'  and  43°  10';  and  sighting  P,  the  angles  of 
depression  taken  in  the  same  order  are  32°  18'  and  21°  42'.  The 
distance  between  the  stations  being  1200  feet;  required  the 
difference  in  height  of  P,  L,  and  N. 


134 


PLANE   SURVEYING. 


2.  To  find  the  height  of  an  object,  PO,  standing  on  the  edge 
of  a  lake  and  inaccessible  to  L,  a 
station  on  the  opposite  rocky 
shore,  a  distance  of  500  feet 
was  measured  from  L  up  the 
slope  to  N,  where  the  angles  of 
depression  of  L,  0,  and  P  were 
observed  respectively,  39°  40', 
25°  20',  and  21°  32'.  Required  the  height  of  PO. 

194.  To  determine  the  height  of  an  object,  and  its  distance 
from  three  observing-stations  situated  in  a 
straight  line  and  in  the  horizontal  plane 
through  the  foot  of  the  object. 

Let  PO  represent  the  required  height ;  L, 
R,  and  N"  the  stations  ;  the  angles  of  eleva- 
tion of  P  taken  at  each  and  in  the  order 
named  a,  ft,  and  0.  The  distance  LR  =  a, 
RN=  b,  and  the  unknown  height  =  x.  It  is 
evident  that  the  triangles  POL,  POR,  and 
PON  are  right-angled  at  0,  and  therefore 

OL  =  x  x  cot  a. 
OR  =  x  x  cot  ft. 
ON=  x  x  cot  6. 

Again,  drawing  OM  perpendicular  to  LN,  we  shall  have  from 
the  acute-angled  triangle  LOR, 


and  from  the  obtuse-angled  triangle  NOR, 


RM; 

or,  substituting  the  proper  values  for  the  lines  represented,  we 

shall  have 

a?  cot2  a  =  a?  cot2  (3  +  a2  -  2  a  MR, 
a?  cot2  0  =  x-  cot2  ft  +  &2  +  2  b  MR. 


INACCESSIBLE   DISTANCES. 


135 


Eliminating  MR  by  multiplying  the  first  by  6,  the  second  by  a, 
adding  and  factoring,  we  obtain 


Whence 


y?  cot2/?  (a  +  &)  +  ab(a  +  6)  . 


ib  cot2  a.  +  acot20  —  cot2/3(a  +  6) 
If  the  stations  are  equidistant,  the  formula  reduces  to 


2a2 


Or, 


Having  obtained  the  height  of  P  above  the  plane,  the  hori- 
zontal distance  from  the  object  to  either  station  may  be  deter- 
mined by  multiplying  this  height  by  the  cotangent  of  the  angle 
of  elevation  at  the  station.  The  oblique  distance  from  either 
station  to  P  is  given  by  the  product  of  PO  and  the  cosecant  of 
the  angle  of  elevation  at  the  station. 

INACCESSIBLE  DISTANCES. 

195.  The  distance  apart  of  three  objects,  L,  0  and  N,  in- 
accessible from  P  are  known,  viz. : 
LO  =  2000  feet,  ON=  1800  feet,  and 
.LJV=2400  feet.  At  P,  situated  in 
the  prolongation  of  ON,  the  observed 
angle  =21°  48';  how  far  is  it  from 
station  P  to  each  object? 

First  calculate  angle  0 ;  then  in  the 
triangle  POL  there  will  be  known  all 
the  angles  and  one  side,  whence  the  re- 
quired distances  may  be  readily  found. 

Usually  the  station  P  cannot  be  chosen  so  as  to  fall  in  ON 
or  OL  produced ;  then  the  measurement  of  two  angles  will 
generally  be  sufficient,  with  the  known  distances  to  locate  the 


136 


PLANE   SURVEYING. 


point  of  observation.     For  example,  suppose  the  distances  and 
angles  are  as  follows : 

NO  =  I  =  3000  feet ; 

OL  =  n  =  3600  feet ; 

LN=  o  =  4800  feet ; 
angle  NPO  =  a  =  23°  40' ; 
angle  LPO  =  (3=22°  50'. 

By  construction,  the  point  P  may  be  found  as  follows :  Sub- 
tract from  180°  2  LPO,  and  from  LO  lay  off  at  L  and  0  the 
angles  LOM  and  OLM,  each  equal  to  half  the  remainder. 
From  the  point  M  thus  determined  as  a  centre,  and  with  a 


radius  LM,  describe  the  circumference  OLP.  The  angle  LPO 
will  then  be  contained  in  the  segment  LPO,  and  the  point  P 
must  be  somewhere  in  the  circumference  OLP.  In  like  man- 
ner, by  means  of  the  angle  OPN,  find  another  circumference 
ONP,  in  which  the  point  P  must  be  situated.  The  intersec- 
tion of  these  circumferences  indicates  its  position. 

The  angle  at  the  circumference  being  half  that  at  the  centre, 
the  angle  LMO,  subtended  by  the  same  chord  as  LPO,  will  be 
•2  LPO,  and  the  angles  OLM  and  LOM  being  equal  and  to- 
gether the  supplement  of  LMO,  each  angle  will 


INACCESSIBLE  DISTANCES. 


137 


Otherwise.  Construct  an  angle  NLR  equal  to  OPN\  also 
LNR  equal  to  OPL,  and  describe  a  circumference  through  the 
points  L,  It,  and  N.  The  point  P  must  lie  in  the  circumfer- 
ence, and  also  in  the  line  drawn  from  0 
through  R.  Their  point  of  intersection 
therefore  will  indicate  its  position. 

The  student  may  give  the  reason. 

196.  By  Calculation.  Pass  a  circle 
through  the  points  L,  N,  P,  and  join  L 
and  N  with  R,  thus  forming  a  triangle 
in  which  the  angles  RLN  and  RNL  are 
equal,  respectively,  to  the  observed  an- 
gles RPN  and  RPL,  and  these,  with 
the  known  side  LN,  furnish  data  sufficient  to  compute  the  sides 
LR  and  RN.  Next  calculate  the  angle  ONL,  whence,  by  sub- 
traction, the  angle  ONR  is  found.  Now,  in  the  triangle  NOR 
there  are  given  two  sides  and  the  included  angle  to  find  NOR 
and  ORN,  or  its  supplement  PRN,  and  by  means  of  the  sine 
proportion  and  the  triangles  PON  and  POL  the  distances  PN, 
PO,  and  PL  may  be  obtained. 

Otherwise.  After  finding  the  angle  0,  obtain  an  expression 
for  either  OLP  or  ONP,  and  then,  by  the  sine  proportion,  the 
required  distances. 

Denote  the  angle  OLP  by  <£,  ONP  by  i/r,  and  the  other  parts 
as  before  ;  then 


sin  B  :  sin  <k  =  n  :  OP,  or  QP=w 


sin/3 


Whence 


sin  «  :  sin  ^  =  I  :  OP,  or  OP  = 


sin  /8 


sin  a 


and 


n  sin  a 


138  PLANE   SURVEYING. 

Again,     <£  =  360  —  a  —  /?  —  O  —  f; 
or,  putting  360  —  a  —  /3  —  0  =  0, 

<}>=*0  —  fa  in  which  6  is  known  ; 

and 


Developing  the  left-hand  member,  dividing  through  by  cos  ^, 
and  simplifying,  there  results 


tan  *  =  _  ; 

I  sin  0  +  n  sin  a  cos  6 


or,  cot./r  =      .       ^      +  oot0. 

n  sin  a  sin  0 

There  are  therefore  but  three  steps  in  the  solution : 

1.  Calculate  the  angle  0,  and  thence  obtain  0. 

2.  Find  tan  fa  or  cot  fa 

3.  By  sine  proportion,  calculate  PN,  P0,  and  PL. 

In  the  example  given,  since  the  sides  are  in  the  proportion 
5:6:8,  the  angle  0  may  be  readily  found  from  the  well-known 
formula  for  the  cosine  of  an  angle, 

cos  0  =  25+36-64 _ _  Q5 _  ^0  ^ 

uU 


and 

0=213°  38'; 

whence 

*=109°53', 

<£  =  103°  45'. 

sin  23°  40' 

Ar.  co.  =  0.396406 

:  sin  109°  53' 

=  9.973307 

:  :  3000 

=  3.477121 

:  PO  =  7028 

=  3.846834 

*  Regard  must  be  given  to  the  signs  of  the  trigonometrical  functions. 


INACCESSIBLE   DISTANCES. 


139 


sin  23°  40' 
sin  46°  27' 
3000 


Ar.  co.  =  0.396406 
=  9.860202 
=  3.477121 


sin  29°  50' 
sin  46°  25' 
3600 

PL  =  5242 


=  3.733729 

Ar.  co.  =  0.303225 
=  9.859962 
=  3.556303 


=  3.719490 


If  the  supplement  of  the  observed  angles  at  P  equals  the  angle 
at  0,  the  circle  will  pass  through  the  three  points  L,  N,  and  0, 
and  P  may  be  anywhere  on  the  circumference,  and  hence  its 
distance  is  indeterminate  by  the  first  method  given  above  ;  and, 
substituting  in  the  formula  the  proper  values  to  find  coti/r  by  the 
second  method,  the  numerator  of  the  fraction  will  become  in- 
finite, as  also  the  cot0 ;  hence,  such  an  observation  will  fail  in 
both  cases  to  locate  the  point  P. 


EXAMPLE. 

Suppose  £^=960  rods,  NO  576  rods,  LO  640  rods,  the 
angle  LPO=19°,  and  NPO  =  25°.  Find  the  distances  PO, 
PN,  and  PL. 

Ans.    PL  =  758  rods  ;  PO  =  1310  rods  ;  PN=  1350  rods. 

197.    From  the  top  of  a  mountain  m  miles  high  the  angle  of 
depression  of  a  line  tangent  to  the  earth's 
surface  is  a  degrees  ;  it  is  required  thence 
to  find  an  expression  for  the  radius  of 
the  earth,  assuming  it  to  be  a  sphere. 

Let  O  represent  the  centre  of  the 
earth  ;  N  the  mountain  top  ;  P  the  point 
of  tan  gen  cy  ;  OP  and  OK  radii  of  the 
earth;  /22V" the  height  of  mountain  and 
prolongation  of  OR. 


140  PLANE    SURVEYING. 

Draw  NL  perpendicular  to  ON,  and  denote  the  radius  of  the 
earth  by  r ;  then,  since  NL  and  NP  are  respectively  perpendic- 
ular to  NO  and  OP,  the  angle  NOP  =  the  angle  of  depression 
LNP=a. 

Hence  (r  +  m)  cos  a  =  r. 

m  cos  a 


1  —  cos  a 


Ans. 


MISCELLANEOUS  PROBLEMS. 

1.  Determine  the  height  of  a  hill,  knowing  that  the  angle  of 
elevation   of  its   top  from  a  certain   station  =  50°,   and  at  a 
station  800  feet  more  remote  the  angle  of  elevation  =  36°  20'. 

2.  The  angle  of  depression,  taken  from  a  balloon  to  a  station 
whose  horizontal  distance  is  known  =  18°  40'.    Find  the  height 
of  the  balloon. 

3.  Two  war   vessels,  desiring  to  ascertain   their   distances 
from  a  fort,  remove  from  each  other  2000  feet,  anil  measure  the 
angle    between    each   other   and   the   fort ;    the   angles   being 
79°  40'  and  82°  20',  what  were  their  distances? 

4.  Two  observers  on  the  same  horizontal  plane,  1500  feet 
apart,  and  in  a  vertical  plane  with  a  balloon,  observe  its  angles 
of  elevation  to  be  62°  40'  and  71°  10'.     Required  the  height  of 
the  balloon. 

5.  The  passage  between  two  objects  L  and  N  being  ob- 
structed by  a  swamp,  the  lines  £P=420  feet,  and  PN=  540 
feet,  were  measured,  and  the  angle  LPN  observed  =  86°  42'. 
Find  the  distance  LN. 

6.  What  distance  can  a  person  whose  eye  is  5|  feet  above 
the  ocean  see  its  surface?    Assume  radius  =  3960  miles. 

7.  If  the  sun  subtend  an  angle  of  32'  2",  and  his  distance  from 
the  earth  is  93,000,000  miles,  what  is  his  diameter? 

8.  What  is  the  altitude  of  the  sun  when  the  shadow  of  a  staff 
cast  on  a  horizontal  plane  is  to  the  height  of  the  staff  as  7  to  5  ? 


MISCELLANEOUS    PROBLEMS.  141 

9.  If  the  horizontal  parallax  *  of  the  moon  be  56'  50"  and 
the  diameter  of  the  earth  7920  miles,  what  is  the  distance  of 
the  moon  from  the  earth? 

10.  If  the  moon  subtend  an  angle  of  31'  14",  when  its  dis- 
tance is  240,000  miles,  what  is  its  diameter? 

11.  When  the  meridian  altitude  of  the  sun  is  50°,  the  shadow 
cast  by  the  peak  of  a  mountain  reaches  a  certain  point  on  a 
horizontal   plain ;   but  when    his  meridian  altitude    is  60°,  the 
shadow  strikes  a  point  2000  feet  nearer  the  base  of  the  moun- 
tain.    Determine  the  height  of  the  mountain  above  the  plain. 

QUERIES.  If  on  the  same  day  two  observations  were  made 
on  the  sun  for  altitude,  one  or  both  when  he  was  not  on  the 
meridian,  and  the  length  of  the  shadow  measured  as  in  Ex.  11, 
would  sufficient  data  be  thus  obtained  to  determine  the  height 
of  the  mountain  ? 

Would  it  be  possible  with  data  obtained,  as  in  the  first  query, 
to  ascertain  the  height  of  the  mountain  if  the  sun  was  vertical 
over  the  mountain  at  noon  ? 

12.  If  the  height  of  a  mountain  is  m  miles  and  its  top  is  visi- 
ble d  miles,  find  an  expression  for  the  diameter  of  the  earth, 
assuming  it  to  be  a  sphere. 

13.  The  angle  of  depression  taken  on  the  top  of  Peak  of 
Teneriffe,  which  is  two  and  a  half  miles  high,  to  the  farthest 
visible  point  was  2°  2'.     It  is  required  to  determine  the  observed 
distance  and  the  diameter  of  the  earth,  assuming  it  to  be  n 
sphere.  Dist.,  140,876  miles;  Diam.,  7936  miles.   Ans. 


EXERCISES. 

1.    Measure  the  height  of  a  flagstaff  or  church  spire  above 
the  street. 

*  The  angle  at  the  moon,  or  other  heavenly  body,  subtended  by  the 
semi-diameter  of  the  earth. 


142 


PLANE   SURVEYING. 


2.  Measure  the  height  of  a  monument,  tower,  or  some  other 
prominent  building  upon  a  hill,  without  obtaining  the  distance 
to  the  foot  of  the  object.  Also,  if  practicable,  measure  the 
distance  to  the  foot  of  the  object  and  the  proper  angles.  Com- 
pute and  compare  results  with  each  other,  and  with  the  actual 
height,  if  it  can  be  ascertained. 


SECTION  V. 

RECORDING   THE  FIELD   NOTES. 

198.  The  Field  Notes  may  be  recorded  in  various  ways,  de- 
pending upon  the  instrument  used,  and  the  extent  and  intricacy 
of  the  survey. 

First  Method.  If  the  compass  is  employed,  the  bearings 
simply  to  be  taken,  distances  measured,  and  the  tract  bounded 
by  straight  lines  (no  offsets) ,  the  simplest,  most  compact,  and 
also  most  convenient  form  for  the  subsequent  calculation  of  the 
area  is  to  write  the  stations,  bearings,  and  distances  in  three 
columns,  thus : 


STATIONS. 

BEARINGS. 

DISTANCES. 

REMARKS. 

1 

S.  20°  53'  E. 

13.11 

To  a  maple. 

2 

N.  48°  10'  E. 

13.62 

"    birch. 

3 

N.  43°  40'  W. 

4.73 

"    stake  and  stones. 

4 

N.  45°  08'  W. 

4.75 

"    white  oak. 

5 

S.  51J°  W. 

2.53 

"    sandstone. 

6 

S.  72  J°  W. 

6.56 

"    red  oak,  beginning. 

199.  Second  Method.  If  the  tract  is  not  large,  and  there  are 
offsets  in  addition  to  the  bearings  and  distances,  or  if  simply 
the  angles  and  distances  are  measured,  a  very  good  method, 
especially  for  a  beginner,  is  to  make  a  rough  plat  of  the  survey, 


RECORDING    THE   FIELD   NOTES. 


143 


and  indicate  in  their  corresponding  places  on  the  sketch  the 
bearings,  or  angles,  and  the  lengths  of  the  lines  and  offsets,  as 
shown  below : 

6.09, 


The  above  is  a  sketch  of  a  small  field,  showing  offsets  to 
stream,  etc.     The  following  are  hasty  surveys  of  boundaries, 
etc.,  of  land  for  proposed  park  in  City  of  Wilmington,  Del., 
July,  August,  and  September,  1885  : 
Instruments  :  Transit.     Chesterman's  100-foot  steel  tape. 
Work: 

Lines  run  with  transit,  and  carefully  measured  with  steel  tape  from 

station  to  station. 

Angles  between  these  lines  taken,  always  from  left  to  right. 
Magnetic  bearings  of  lines  taken. 
Stations  numbered  or  lettered  in  regular  order. 

Offsets  (sometimes  angles  and  distances)  taken  to  locate  houses,  cor- 
ners of  fences,  etc.,  offsets  made  at  right  angles  with  lines  joining 
stations. 
Notes  : 

Taken  free-hand  in  small  note-books  (size  5;}"  X  3J"). 

Sketches  made  to  suit  the  page  and  to  make  the  matter  clear  for 

plotting. 

The  usual  checks  used  on  field  and  office  work. 
Explanation  of  Sketches  : 

No.  1.   Single  page  of  note-book.     Location  of  fences  on  boundary  of 

land  proposed  for  park. 
No.  2.   Two  opposite  pages  of  note-book.     Location  of  road  through 

land  proposed  for  park,  showing  railroad  crossing. 
No.  3.   Two  opposite  pages  of  note-book.    Location  of  run  between  two 

adjoining  owners  of  land  proposed  for  park. 

No.  4.   Two  opposite  pages  of  note-book.     Location  of  houses,  etc.,  in 
land  proposed  for  park. 


144 


PLANE  SURVEYING. 


42V 


^x 

—  •—  • 

/ 

\      -V*! 

ssyir* 

jt-     - 

fvp 

i  • 

s 

\ 

1 

j: 
"a 
§ 

§ 

S 

Static 

iPost 


No.  1. 


RECORDING   THE  FIELD   NOTES. 


145 


Sta.  R5 


107V 


No.  2. 


146 


PLANE   SURVEYING. 


KECOKDLN'G    THE  FIELD   NOTES. 


147 


177  2i- 


200.  Third  Method. 
The  column  method,  an- 
alogous to  that  shown 
in  Article  38,  Chain 
Surveying,  is,  however, 
the  most  general.  If 
the  bearings  are  taken, 
they  may  be  inserted  in 
the  column  either  verti- 
cally or  diagonally  ;  if 
only  the  angles  are  ob- 
served, they  should  be 
placed  at  the  stations 
which  indicate  where 
the  measurements  were 
made.  The  objects  to 
which  offsets  are  meas- 
ured may  be  designated 
or  delineated  on  the 
marginal  side  of  the  line 
as  they  naturally  ap- 
pear. Where  streams, 
roads,  fences,  etc.,  cross 
the  line,  representations 
of  them  are  made,  in- 
dicating approximately 
their  direction ;  or,  if 
desirable,  their  bear- 
ings, or  angular  devia- 
tions from  the  line,  may 
be  taken  and  recorded. 

The  following  notes 
will  more  fully  explain 
the  method  under  con- 
sideration : 


No. 


148 


PLANE   SURVEYING. 


24.28 

21.08 
20.68 


13.50 
6.00 


(6) 

11.38 


CO 

(5) 


(4) 

17.54 

M 

a 

CO 

f3) 
33.10 
24.75 


(2) 
11.90 


0) 


Chestnut  stump,  cor.  to  J.S.K.  in  W.D.C. 
line. 


West 


Stone,  7  links  S.E.  of  a  butternut  tree 
Cor.  to  A.L.  &  W.D.C. 


Limestone,  middle  of  public  road. 


Stake  &  Stones,  cor.  taL.R.  in  A.L.'s  line 
Middle  of  road. 


White  oak  stump  10'  S.W.  stone  house, 
north  side  public  road, cor.  to  J.V.  &  J.L. 


RECORDING   THE   FIELD   NOTES. 


149 


» 

(i) 

H3.3Q 

—  =%=S:=SP=5= 

14.52 

g-—  5-^=^=^ 

_—  •  —  -_  •— 

14.12 

.  .  _,  ^^l^^r-^J'ftSt. 

0 

fc 

(12) 

Limestone,  cor.  to  W.V.  in  C.C.'s  line 

10.47 

, 

0 

Oi 

00 
CO 

(11) 

Limestone. 

8.00 

W 

» 

fc' 

! 

(10) 

Limestone,  cor.  to  J.S.K.  in  C.C.'s  line 

30.60 

CO 

(9) 

Limestone. 

8.31 

CO 

(8) 

Limestone. 

4.45 

8 

i 

(7) 

150 


PLANE   SURVEYING. 


k 

(4) 

Limestone.    Sta.  (4)  in  foregoing 

10.49 

description. 

Stone  house    |p 

9.00 

100'  from  line. 

ri 

§0 

oo 

02 

m 

Limestone. 

N.  12°  W. 

10.41 

Lane  leading  to  dwelling,  S.  11°  E. 

Road 

9.00 

Hf  Barn. 

| 

i 

6.40 

M 

j 

3 

House  60'  from  line. 

-1 

W 

0 

8 

. 

20.38 

0 

8 

Stone  house   Wft, 

(i) 

White  oak  stump.     Sta.  (1)  :u 

f           near  corner. 

foregoing  description. 

RECORDING   THE   FIELD   NOTES. 


151 


The  bearing  and  distance  of  proof-line  from  P  to  Station  (11)=  S.62£°  W. 
19.10. 


N.  79°  10'  E.  6.00 


Housi 
o  Spring  Run 


and  15.82toEastlineof  Survey 


9.20 


5.75 


OD 

(#) 

8.80 

W 

02 


(I/) 

36.20 


22.38 

W 

02 


Limestone  at  end  of  lane  on 
north  bank  of  "  Big  Brook." 


Limestone  on  bank  of  Spring 
Run. 


Point  in  lane. 


Limestone  in  middle  of  public 
road  at  end  of  lane. 


152  PLANE  "SURVEYING. 

The  notes  show  that  the  sides  of  the  tract  were  first  surveyed ; 
which,  with  their  bearings  and  distances,  include  also  the  loca- 
tion and  general  direction  of  road-crossings,  streams,  etc.,  a 
description  of  the  corners,  and  the  names  of  owners  of  property 
adjoining  the  survey.  Next  to  traversing  the  bounding  lines, 
the  survey  of  the  public  road,  crossing  the  farm  from  east  to 
west,  was  made.  This  road  enters  the  tract  at  station  (1)  ; 
at  6.40  chains  from  (B)  it  passes  a  house  which  is  60  feet 
to  the  right;  at  9.00  chains  a  road  to  the  left,  the  bearing  of 
which  is  given  ;  at  10.41  chains  is  a  corner  at  end  of  lane  lead- 
ing to  dwelling ;  near  the  east  end  of  road  a  stone  house  is 
located,  at  100  feet  north  of  the  line  ;  and  at  10.49,  station  (4) 
of  sides  survey  is  reached,  at  which  point  the  road  leaves  the 
farm.  The  survey  of  the  lane  to  the  dwelling,  and  thence  to 
the  creek,  is  next  recorded.  Here  are  noted  the  intersection  of 
a  line  S.  79°  10'  W.,  and  the  distances  on  this,  east  and  west, 
to  spring  runs,  as  well  as  the  distances  to  the  east  and  west 
sides  of  the  tract;*  the  dwelling  and  barn  are  located,  and  the 
limestone  on  the  north  bank  of  Big  Brook  reached.  A  line  was 
run  from  this  last  point  to  station  (11),  which,  in  connection 
with  the  survey  of  the  lanes,  the  public  road,  and  the  cross-line 
from  L  to  F,  gave  proof  of  the  accuracy  of  the  work. 

*  This  line  was  made  a  boundary  in  the  subsequent  division  of  the  land. 


KECOKDING   THE   FIELD   NOTES. 


153 


154  PLANE   SURVEYING. 

SECTION  VI. 

LATITUDES  AND  DEPARTURES. 

201.  The  Difference  of  Latitude  of  the  two  ends  of  a  line  is 
the  perpendicular  distance  between  the  parallels  of  latitude 
which  pass  through  them,  and  is  reckoned  north  or  south, 
according  as  the  bearing  is  northerly  or  southerly. 

The  Difference  of  Longitude  of  the  two  ends  of  a  line  is  the 
perpendicular  distance  between  the  meridians  which  pass  through 
them,  and  is  reckoned  east  or  west,  according  as  the  bearing  is 
easterly  or  westerly. 

The  difference  of  latitude  of  a  line  is  often  called  briefly  the 
latitude,  or  northing  or  southing  ;  and  the  difference  of  longitude, 
its  departure,  or  easting  or  westing. 

It  will  be  perceived  from  the  definitions  just  given  that,  when 
a  line  bears  either  due  north  or  south,  the  distance  equals  the 
latitude,  and  the  departure  is  nothing ;  but  if  the  bearing  is 
east  or  west,  the  distance  and  departure  are  equal,  and  the  lati- 
tude is  zero.  Furthermore,  it  will  be  seen  that  in  all  other 
cases  except  those  just  cited,  the  latitude,  departure,  and  dis- 
tance form  the  three  sides  of  a  right  triangle  :  the  distance 
being  the  hypotenuse,  and  the  latitude  and  departure  the  sides 
about  the  right  angle. 

Let  LP  represent  a  line  given  by  its  bearing  and  distance  ;  it 
is  required  to  determine  its  latitude  and  departure. 

Let  OL  and  PM  represent  parallels 
of  latitude,  and  LM  and  OP  meridians. 
The  lengths  of  LM  =  OP  and  LO=MP 
are  required. 

The  problem  stated  simply  is  :  Given 
in  a  right  triangle  LMP  the  hypotenuse 
LP  (distance),  the  angle  L  (bearing), 
to  find  the  side  LM  (latitude) ,  and  MP 
(departure). 


LATITUDES    AND    DEPARTURES. 
From  Trigonometry,  LM^  LPvosL, 


155 


So  it  is  seen  that  the  latitude  of  a  line  is  obtained  by  taking 
the  product  of  the  distance  and  the  cosine  of  the  bearing,  and 
the  departure  is  equal  to  the  product  of  the  distance  and  sine  of 
the  bearing. 

202.  The  case  just  treated  is  the  principal  one  which  the 
surveyor  will  use,  since  it  is  necessary  —  as  will  subsequently  be 
seen  —  in  computing  areas,  to  determine  the  latitudes  and  de- 
partures ;  and  by  Jthese  formulas  he  will  generally  obtain  them, 
having  taken  in  the  field  the  bearings,  or  angles,  and  distances. 

Other  cases,  however,  will  occur  in  practice  referring  to  the 
triangle  LMP,  and  for  convenience  they  are  here  subjoined. 

Designating  the  length  of  the  line,  or  distance,  by  s,  the  bear- 
ing by  6,  the  latitude  and  departure  respectively  by  I  and  d, 
then  we  may  write  the  following  formulas : 


CASE. 

GIVEN. 

KEQUIKED. 

FORMULAS. 

1 

b,      s. 

/,          d. 

/  =  s  cos  b,                  d  =  s  sin  b. 

2 

b,      I. 

s,         d. 

s  =  —  =  lsecb,     d  =  l  tan  b. 
cos  b 

3 

b,     d. 

s,          I. 

sin  6                             tan  6 

4 
5 

s,      I. 
s,     d. 

b,         d. 
b,          I. 

cos  6  =  -,                   d  =  Vs'2  —  I'2, 
s 

sin  6   =-,                    /=  V*3  —  rf». 

s 

6 

I,      d. 

b,         s. 

tan  b  =  -,                     s=  VP  +  rf*. 

EXAMPLES. 

1.    Given  the  bearing  and  distance  of  a  line,  N.  23°  54'  W. 
18.25  chains  ;  required  its  latitude  and  departure. 


156  PLANE   SU It V EYING. 

2.  Given  the  bearing  of  a  line  Jf .  87°  40'  E.,  and  the  depart- 
ure 2640  feet ;  find  its  distance  and  latitude. 

3.  Given  the  length  of  a  line  24.60  chains,  and  the  departure 
17.40  ;  find  its  bearing  and  latitude. 

4.  Given  the  latitude  23.76  chains  south,  and  the  departure 
0.94  chains  west;  required  the  bearing  and  distance. 

5.  Given  the  distance  1886  feet,  and  the  latitude  943  ;  deter- 
mine its  bearing  and  departure. 

6.  It  is  required  to  find  the  distance  and  departure  of  a  line, 
given  the  bearing  S.  30'  W.,  and  latitude  10.80  chains. 

203.  The  Traverse  Table.  By  the  use  of  Formula  1,  last 
article,  latitudes  and  departures  have  been  calculated  for  every 
quarter-degree  of  the  quadrant,  corresponding  to  distances  from 
1  to  10,  and  even  from  1  to  100 ;  these  results  tabulated  con- 
stitute the  traverse  table.  Such  a  table  was  considered  quite 
indispensable  when  the  compass  was  the  principal  surveying 
instrument,  but  since  the  more  accurate  transit  has  to  a  great 
extent  superseded  the  compass,  and  surveyors  are  now  reading 
to  minutes  instead  of  quarter-degrees,  the  common  traverse  table 
reading  only  to  quarter-degrees  is  of  little  practical  value. 

When,  therefore,  the  bearings  are  read  to  minutes,  the  lati- 
tudes and  departures  are  generally  best  obtained  from  a  table 
of  natural  sines  and  cosines.* 

However,  for  the  benefit  of  those  engaged  in  compass  survey- 
ing, and  for  those  who,  though  reading  to  minutes,  prefer  to 
obtain  by  interpolation  the  latitudes  and  departures  from  the 
traverse  table,  one  is  given  near  the  end  of  this  volume. 

*  A  traverse  table  in  which  the  calculations  are  made  to  every  minute 
of  bearing  for  distances  from  1  to  10  and  extending  to  five  decimal  places, 
would  answer  the  purpose  admirably.  Such  a  table  is  in  existence,  but  it 
is  not  common.  The  common  tables  of  natural  sines  and  cosines  are  sim- 
ply tables  of  latitudes"  and  departures  corresponding  to  a  unit's  distance. 
With  a  distance  2,  the  latitude  and  departure  are  twice  those  in  the  table; 
when  the  distance  is  3,  three  times  ;  when  n,  n  times. 


LATITUDES    AND   DEPARTURES.  157 

Explanation  of  the  Traverse  Table.  The  number  of  degrees 
in  the  bearing  if  it  does  not  exceed  45  is  found  in  the  left-hand 
column  of  the  page,  and  the  latitudes  and  departures,  as  indi- 
cated at  the  top,  may  be  taken  under  the  proper  distance ;  if 
the  number  of  degrees  is  greater  than  45,  it  is  found  in  the  right- 
hand  column  of  the  page,  and  the  columns  of  latitudes  and 
departures  are  indicated  at  the  bottom.  For  example  : 

1.  Let  it  be  required  to  find  the  latitude  and  departure  cor- 
responding to  a  bearing  N.  34°  30'  E.  and  distance  5  chains. 

We  find  in  the  table,  opposite  34°  30'  and  under  "distance  5," 
in  the  column  headed  "Lat.,"  4.121,  and  in  the  column  headed 
"Dep.,"  2.832.  Hence  the  latitude  and  departure  are  respec- 
tively 4.12  N.  and  2.83  E. 

2.  Required  the  latitude  and  departure  of  a  line  bearing  N. 
72i°  W.  9  chains. 

Looking  in  the  column  at  the  right  of  the  page  for  72°  15', 
and  under  "distance  9,"  we  find,  reading  at  bottom, 
in  the  Lat.  column,  2.744  ; 
in  the  Dep.  column,  8.572. 

Hence  the  latitude  is  2.74  chains  N.,  and  the  departure  8.57 
chains  W. 

204.  The  table  may  be  used  to  find  the  latitude  and  depart- 
ure for  any  distance  however  great.  If,  in  first  example  above, 
we  suppose  the  bearing  to  remain  the  same,  but  the  distance  to 
be  50  chains  ;  then,  since  for  the  same  bearing  the  latitudes  and 
departures  vary  directly  as  the  distances,  the  latitude,  or  depart- 
ure, for  50  chains  is  10  times  that  for  5  ;  and,  as  multiplying 
by  10  is  in  effect  removing  the  decimal  point  one  place  to  the 
right,  we  may  take  directly  from  the  table  opposite  5  the  lati- 
tude and  departure  of  50,  or  41.21  N.  and  28.32  E. 

If  the  distance  is  not  a  multiple  of  10,  but  made  up  of  units 
and  tens,  we  may  take  out  of  the  table  the  latitude  and  depart- 
ure for  the  units,  and  for  the  tens  as  indicated  above.  The  sum 
of  these  will  evidently  be  the  latitude  and  departure  required. 


158  PLANE  SURVEYING. 

3.  Let  it  be  required  to  find  the  latitude  and  departure  of  a 
line  S.  40°  E.  34  chains. 

Looking  in  the  table  opposite  40°  and  under  "  distance  3," 
take  out  at  ouce,  by  conceiving  the  decimal  point  removed  one 
place  to  the  right. 

For  30  chains,         Lat.  22.98         Dep.  19.28 
Then     "     4      "  "       3.06  "       2.57 

34  chains,         Lat.  26.04  S.   Dep.  21.85  E. 

By  an  extension  of  the  above  principle,  the  table  may  be  used 
to  obtain  the  latitude  and  departure  when  the  distance  is  com- 
posed of  chains  and  links. 

4.  Given  the  bearing  of  a  line  S.  28°  45'  W.  26.58  chains,  to 
find  its  latitude  and  departure. 

For  20       chains,       Lat.  =  17.53  Dep.  =    9.62 


6 
.5        " 
•  .08      « 

44    =    5.26 
"    =      .44 
44    =      .07 

44    =    2.89 
44    =      .24 
4'    =      .04 

26.58  chains, 

Lat,  =  23.308. 

Dep.  =  12.  79  W. 

5.  Find  by  the  traverse  table  the  latitude  and  departure  of  a 
line  bearing  N.  41°  45'  E.  17.29  chains. 

6.  Given  the  bearing  of  a  line  S.  £°  W.,  distance  23.48 
chains,  to  find  its  latitude  and  departure. 

7.  What  are  the  latitude  and  departure  of  a  line  bearing  S. 
85°30'E.  135.42  chains? 

8.  If  the  bearing  and  distance  are  N.  89f°  W.  20.09  chains, 
what  are  the  latitude  and  departure  ? 

205.  By  means  of  interpolation  the  traverse  table  may  be 
used  to  find  the  latitude  and  departure  when  the  bearing  is 
given  to  minutes.  Thus,  the  bearing  being  N.  34°  20'  E.  any 
given  distance,  take  out  the  latitude  and  departure  corre- 


LATITUDES   AND  DEPARTURES.  159 

spending  to  34°  15'  and  the  given  distance,  and  add*  to  that 
departure  T5^,  or  £,  of  the  difference  between  it  and  that  corre- 
sponding to  34°  30'  and  the  given  distance,  for  the  departure 
required.  Likewise  obtain  T57  of  the  difference  between  the 
latitudes  corresponding  to  34°  15'  and  34°  30'  and  the  distance, 
and  subtract*  from  the  latitude  first  found  for  the  latitude 
required. 

For  a  bearing  34°  23',  the  fractional  part  to  be  taken  of  the 
difference  between  34°  15'  and  34°  30'  would  be  T% ;  the  numer- 
ator being  the  excess  in  minutes  above  the  quarter,  and  the 
denominator  15. 

206.  In  the  absence  of  a  traverse  table  calculated  to  minutes, 
the  table  of  natural  sines  and  cosines,  as  before  stated,  is  the 
best  to  use  when  the  bearings  are  given  to  minutes. 

It  is  shown  in  Article  201  that  the  cosine  of  the  bearing  mul- 
tiplied by  the  distance  gives  the  latitude,  and  the  product  of  the 
distance  and  sine  of  bearing  gives  the  departure. 

EXAMPLES. 

1.  The  bearing  and  distance  of  a  line  are  N.  37°  43'  W. 
24.29  chains ;  required  its  latitude  and  departure. 

Four  places  of  decimals  from  the  table  will  usually  be 
sufficient. 

The  cosine  of  37°  43'  true  to  four  places  =  .7911. 
The     sine  of  37°  43'  true  to  four  places  =  .6118. 

.7911X24.29  =  19.21  N.  Lat. 
.6118  x  24.29  =  14.86  W.  Dep. 

The  following  contracted  form  of  multiplication,  using  five 
decimal  places,  gives  practically  the  same  result : 

Cosine  of  bearing  =  .79105  ;  sine  of  bearing  =  .61176. 


*  The  departure  increases  with  an  increase  of  the  bearing;  the  latitude 
diminishes. 


160  PLANE   SURVEYING. 

chains,     Lat.  =  15.8210       Dep.  =  12.2352 

"    =    3.1642         "      =    2.4470 
Distances 


.0712         "     =      .0551 


24.29  chains,     Lat.  =  19.21  N.     Dep.  =  14.86  W. 

2.  Find  the  latitude  and  departure  of  a  line  bearing  S.  62°  1 7'  E. 
37.18  chains. 

3.  Required  the  latitude  and  departure  of  a  line  N.88°57'W. 
28.97  chains. 

4.  Required  the  latitude  and  departure  of   a   line  bearing 
S.  i°  E.  2640  feet. 

5.  Given  the  bearings  and  distances  of  two  lines  running 
from  the  same  point  P,  as  follows:  PO,  N.  38°  37'  E.  1760 
feet,  and  PL,  N.  71°  54'  E.  1320  feet ;  to  find  by  means  of  lat- 
itudes and  departures  the  distance  OL. 

6.  Assuming  PO bears  N.  48°  17'  W.  27.42  chains,  and  PL 
S.  36°  28'  W.  19.24  chains,  find,  as  in  the  last  example,  the 
distance  OL  between  the  extremities  of  the  lines. 

207.  Testing  a  Survey.  It  is  evident  that  when  a  surveyor 
has  passed  completely  round  a  tract  of  land  and  returned  to  the 
place  of  beginning,  he  has  gone  in  a  northerly  direction  just  as 
far  as  he  has  gone  in  a  southerly  direction,  and  as  far  easterly 
as  westerly.  Hence  the  sum  of  the  north  latitudes  should  equal 
the  sum  of  the  south  latitudes,  and  the  sum  of  the  east  depart- 
ures equal  the  sum  of  the  west  departures.* 

In  practice,  this  degree  of  accurac}-  is  seldom  attained,  for 
various  causes  incident  to  the  manipulation  of  the  instruments, 
their  inherent  defects,  imperfect  chaining,  etc. 

*  If  the  survey  is  effected  by  traversing  (Article  163),  the  reading  at 
the  last  station  should  be  360°  or  0°.  If  the  interior  angles  are  measured, 
their  sum  should  equal  twice  as  many  right  angles  less  four  as  the  figure 
has  sides.  If  a  small  error  exists,  it  must  be  distributed  evenly  among  the 
angles,  unless  on  account  of  the  difficulty  of  observing  one  or  more  of  the. 
angles,  these  should  have  a  larger  share  of  the  error.  See,  also,  Article  156 


LATITUDES    AND   DEPARTURES. 


161 


On  account  of  the  varying  conditions  in  different  surveys,  it 
is  impracticable  to  state  precisely  how  great  an  error  should  be 
allowed  without  a  re-survey  of  the  tract.  A  rule  usually  fol- 
lowed by  compass  surveyors  is  to  allow  an  error  of  1  link  for 
every  5  chains,  1 :  500. 

This  is  perhaps  a  fair  average  for  ordinary  farm  surveying. 
If  the  ground  is  exceptionally  clear,  and  quite  level,  an  error  of 
1  :  1000  is  not  too  great ;  if,  on  the  other  hand,  the  ground  is 
uneven,  rocky,  and  brush}7,  1  :  300,  or  even  1 :  200,  might  be 
allowed.  The  error  resulting  from  a  transit  survey  of  the  same 
ground  should  be  much  less.  For  the  average  case  given  above, 
instead  of  1 :  500  it  should  not  be  much  less  than  1  :  1200. 

The  above  rules  are  cited  simply  as  guides  to  the  young  sur- 
veyor to  aid  him  in  forming  a  standard  for  himself,  based  on 
his  own  experience. 

208.  Correcting  Latitudes  and  Departures,  or  Balancing  the 
Survey.  (1)  A  survey  is  balanced  when  the  northings  equal 
the  southings,  and  the  eastings  equal  the  westings.  When  these 
equalities  do  not  exist,  the  error  is  distributed  among  the  lines, 
proportioned  to  their  lengths.  This  operation  is  called  correct- 
ing the  latitudes  and  departures.  It  is  best  illustrated  by  an 
example : 


•7. 

BEARINGS. 

DISTS. 

LATITUDES. 

DEPART- 
URES. 

CORREC- 
TIONS. 

CORRECTED 
LATITUDES. 

CORRECTED 
DEPART'S. 

f. 

S. 

E. 

W. 

j 

* 

N. 

S. 

E. 

W. 

i 

8.  20°  53'  E. 

13.11 

.  .  . 

12.25 

4.67 

1 

1 

12.27 

4.68 

I 
1 

4 
f 

N.  48°  10'  E. 
N.  43°  40'  W. 
N.  45°  08'  W. 
8.  51°  30'  W. 

13.62 
4.73 
4.75 
2.53 

9.08 
3.42 
3.35 

... 

10.15 

3.26 
3.36 
1.98 

9 

1 
1 

1 
1 

9.06 
3.41 
3.34 

10.16 

3.26 
3.35 
1.98 

1.57 

... 

1.57 

I 

S.  72°  30'  W. 

6.56 

... 

1.96 

... 

6.26 

7 

1 
4 

... 

1.97 

... 

6.25 

45.30 

15.85 

15.78 

14.82 

14.86 

15.81 

15.81 

14.84 

14.84 

15.78 

14.82 

Error  in  latitude,   7  links.                        4  linkfl.  Error  in  departure. 

162  PLANE   SUliV  EYING. 

In  the  table  the  latitudes  and  departures  corresponding  to  the 
several  bearings  and  distances  are  obtained  by  means  of  a  table 
of  sines  and  cosines,  and  placed  in  their  proper  columns. 

The  first  course  being  between  the  south  and  east,  the  lati- 
tude found  is  written  in  the  column  headed  S.,  the  departure 
in  column  E.,  and  so  on,  the  letters  of  the  course  indicating  the 
columns  in  which  to  place  the  latitudes  and  departures.  The 
difference  of  the  sums  in  the  latitude  columns  is  then  taken,  and' 
found  to  be  7  links :  this  is  the  error  in  latitude. 

The  error  in  departure,  found  in  a  corresponding  manner,  is 
4  links. 

The  total  distance  round  the  field  is  showYi  by  the  footing  of 
the  distance  column  to  be  45.30  chains.  The  distribution  of  the 
error  is  effected  then  by  the  proportions : 

For  the  Latitude. 

Sum  of  the  sides  :  length  of  any  side  =  error  :  correction  for  that  side. 
45.30        :  13.11          =    7     :  2 

45.30        :  13.62          =    7    :  2 

For  the  Departure. 
45.30        :  13.11          =    4    :  1 

It  is  unnecessary  usually  to  make  but  one  proportion  each  for 
the  latitude  and  departure  correction,  since  the  error  for  any 
other  side  may  be  found  mentally  by  comparing  its  length  with 
that  of  the  side  used  in  the  proportion.  Whole  links  only  are 
used.  The  latitude  correction  for  the  second  side  is  a  little 
greater  than  2,  but  it  is  nearer  2  than  3,  and  is  therefore 
called  2. 

The  corrections  thus  found  are  written  in  their  proper  col- 
umns, headed  "Correction,  Lat.  Dep.,"  opposite  the  sides  to 
which  they  refer,  and  are  so  applied  by  addition  or  subtraction 
as  may  be  required  to  reduce  the  errors  to  zero.  The  quanti- 
ties thus  obtained  are  placed  in  the  columns  of  corrected  lati- 
tudes and  departures  to  the  right  of  the  corrections. 


LATITUDES    AND   DEPARTURES.  163 

Since  the  southings  are  too  small,  the  correction  2  is  added 
to  12.25,  making  12.27,  for  the  first  entry  in  the  column  of  cor- 
rected latitudes.  The  eastings  being  too  small,  the  correction 
1  is  added  to  4.67,  making  4.68,  to  be  written  under  E.  in  the 
corrected  departures  ;  and  so  on  for  the  rest. 

If  the  corrections  have  been  properly  applied,  the  northings 
will  equal  the  southings,  and  the  eastings  the  westings,  and  the 
survey  is  balanced. 

In  the  example  just  given,  the  difference  of  latitude  is  7  and 
the  departure  4  links ;  hence,  the  length  of  a  line  to  close  the 
survey  =  A/72  +  4*  =  about  8  links  ;  and  as  the  perimeter  of  the 
tract  =45. 30  chains,  the  "error  of  the  survey,"  or  "  error  of 
closure, "=  1  link  for  5.66  chains,  or  1  :  566. 

Some  surveyors  prefer  a  more  compact  table  than  that  given 
above,  and  instead  of  a  double  set  of  latitudes  and  departures, 
use  but  one,  and  write  in  ink  of  different  colors  the  corrected 
latitudes  and  departures  over  the  first.  Others,  again,  prefer 
two  columns  instead  of  four  for  the  latitudes  and  departures, 
using  the  plus  (-f)  sign  to  indicate  north  latitudes  and  east 
departures,  and  the  minus  (  — )  sign  to  indicate  south  latitudes 
and  west  departures. 

The  form  given  above  is,  however,  preferable  to  either,  since 
a  mistake  in  the  application  of  the  corrections  is  in  that  more 
easily  detected,  the  footings  are  more  expeditiously  and  accu- 
rately obtained,  and  the  subsequent  part  of  the  work  referring 
to  the  area  is  thereby  facilitated. 

If  a  side  of  the  survey  passes  over  very  rough  ground,  or 
through  a  dense  wood,  or  for  any  reason  it  is  rendered  more 
difficult  to  measure  than  any  of  the  others,  the  surveyor  should 
exercise  his  judgment  in  deciding  how  much  more  of  the  error 
than  the  rule  would  indicate  should  be  applied  to  that  side. 

Regard  must  also  be  had  to  the  probability  of  error  in  the 
bearings ;  hence,  when  a  side  of  considerable  length  is  aligned 
through  a  thicket,  or  over  very  uneven  ground,  and  where  often- 
times the  observations  are  made  to  top  of  rod,  if  it  is  found 
that  a  slight  change  in  the  bearing  will  diminish  materially  the 
error,  the  change  should  be  made. 


164  PLANE  SURVEYING. 

The  diurnal  variation  of  the  needle  is  not  unfrequently  a 
source  of  error  in  compass  surveys.  A  range  of  10  minutes  is 
quite  common,  and  even  15  minutes  is  occasionally  noted.  This 
error  may  be  avoided  by  measuring  the  angles  of  the  tract,  or 
testing  the  compass  every  two  or  three  hours  by  setting  up  and 
sighting  on  some  line  as  standard. 

Some  authors  and  surveyors  affirm  that  when  the  bearing  of 
a  line  is  due  east  or  due  west,  the  error  in  latitude  is  nothing, 
and  therefore  such  a  line  needs  no  correction.  Likewise  a  due 
north  and  south  line  has  no  error  in  departure.  The  writer  does 
not  concur  in  this  view  ;  for  the  errors  in  compass  work  are  not 
confined  to  the  chaining,  and  in  transit  surveying  there  is  fre- 
quently considerable  error  in  the  angles.  In  the  application  of 
the  rule  these  facts  are  assumed  ;  indeed,  as  soon  as  a  correc- 
tion, made  in  the  usual  manner,  is  applied  to  any  side,  a  change 
of  bearing  results,  for  the  corrected  latitudes  and  departures  no 
longer  belong  to  the  original  bearing,  but  to  some  other.  More- 
over, there  is  no  more  reason  for  supposing  a  line  runs  due 
north  because  it  is  so  read  than  that  a  line  runs  N.  ^°  E.  or 
N.  89|°  E.  being  so  read  ;  yet  no  surveyor  would  hesitate  to 
apply  the  rule  to  either  of  these,  thus  assuming  that  an  error  in 
bearing  as  well  as  in  chaining  was  committed  ;  and  this  is  the 
correct  assumption  on  which,  without  excepting  any  side,  the 
distribution  of  the  error,  except  as  follows,  should  be  made. 

(2)  If,  however,  a  survey  is  made  with  a  transit  in  good  ad- 
justment, the  angles,  either  interior  or  deflection,  being  carefully 
observed,  and  the  test  hereinbefore  mentioned  when  applied  giv- 
ing the  inference  that  the  angles  were  accurately  measured,  and 
the  error  of  closure  therefore  due  to  erroneous  chaining,  then 
the  correction  which  should  be  applied  is  obtained  as  follows  : 

Add  up  the  columns  of  latitudes,  and  also  those  of  departures, 
and  say,  as  the  arithmetical  sum  of  all  the  -{  *  }•  «*  to 


correction  to  be  applied  to  that  i, 


LATITUDES    AND   DEPARTURES. 


165 


(3)  If  greater  accuracy  is  required  than  can  be  attained  by 
the  preceding  methods,  each  side  should  be  weighted  ;  that  is  to 
say,  the  surveyor  determines  the  relative  difficulties  in  measure- 
ment and  alignment  of  the  boundaries,  considering  some  one 
side  the  standard.  Calling  the  error  probably  made  in  the  side 
chosen  as  standard  one  (1),  another  side,  which  in  the  judg- 
ment of  the  surveyor  was,  per  unit,  twice  as  difficult  to  measure, 
would  be  multiplied  by  2,  or,  as  it  is  termed,  have  a  weight  of 
2  ;  another  multiplied  by  3,  or  1£,  etc.  Then,  instead  of  tak- 
ing the  perimeter  for  the  divisor,  as  was  done  in  the  first  case 
above,  the  sum  of  the  sides  thus  multiplied  or  weighted  is  used, 
and  the  proportion  is  as  follows  : 

As  the  sum  of  the  multiplied  distances  is  to  any  particular 
multiplied  distance,  so  is  the  error  in  •{  l^iltwd*  j.  to  the  correction 

•.  ' 


The  following  illustrates  the  method  of  balancing  a  survey 
when  the  sides  are  weighted  : 


STATIONS. 
BEARINGS. 

DISTANCES. 

WEIGHTS. 

1  MULTIPLIED 
DISTANCES. 

LATITUDES. 

DEPART- 
URES. 

«    § 

<s  ^ 

CORRECTED 
LATITUDES. 

CORRECTED 
DEPART'S. 

JV. 

S. 

E. 

W. 

\ 

1 

jr. 

S. 

E. 

W. 

1  N.  9°  W. 
2  If.  31°  E. 

15.50 

25.40 

1 

8 

15.50 
76.20 

15.31 
21.77 

2.43 

1 

6 

3 
9 

15.32 
21.83 

2.41 

•  . 

13.09 

.  .  . 

13.18 

3  S.  71°  E. 

10.00 

8 

30.00 

3.17 

9.48 

8 

4 

3.14 

9.52 

4  S.  10i°E. 

19.70 

8 

39.40 

19.37 

3.59 

8 

5 

19.34 

3.64 

5  8.  10|°W. 

14.60 

11 

21.90 

14.34 

2.72 

2 

2 

14.32 

2.70 

6  8.  89°  W. 

21.25 

1 

21.25 

0.37 

37.25 

... 

21.25 
26.40 

2 

2 

37.15 

0.35 
37.15 

26.34 

21.23 
26.34 

204.00 

37.08 

26.16 

37.08 

26.16 

Error  in  latitude,  17  links.         24 

inks,  error  In  departure. 

*  Weights  could  be  applied  to  the  correction  of  the  chaining  in  the 
second  case,  by  multiplying  the  latitudes  and  departures  instead  of  the 
lengths  of  the  sides. 


166  PLANE   SURVEYING. 

EXAMPLES. 

Correct  the  latitudes  and  departures  in  the  following  examples 
by  the  first  method : 

1.  2. 

(1)  S.       £°  E.  22.45  chains;  (1)  South  22s45  chains  ; 

(2)  N.  89|°E.  67.10      "  (2)  East    67.10      « 

(3)  N.      i°W.  23.85      "  (3)  North  23.85      « 

(4)  S.  89f°  W.  66.30      "  (4)  West  66.30      " 

(5)  S.  21 1°  W.  1.30      "  (5)  S.  22°  W.  1.30" 

EXERCISES. 

A  few  surveys  should  now  be  made,  and  the  methods  above 
given  employed  in  balancing. 


SECTION  VII. 

SUPPLYING  OMISSIONS. 

209.  When,  for  any  cause,  it  is  impracticable  to  obtain  the 
direction  or  the  length,  or  both,  of  a  side  of  a  tract  of  land, 
these  may  be  obtained  by  calculation.  Even  the  lengths  or 
bearings  of  two  sides  may  in  general  be  supplied.* 

The  determination,  however,  of  these  sides  or  bearings  is 
based  upon  the  measurements  of  the  other  bounding  lines  and 
angles ;  but  as  these  are  not  usually  precisely  correct,  and  as 
there  are  no  means  of  testing  them  in  their  application  to  the 
solution  of  problems  under  this  head,  it  is  earnestly  recom- 
mended that  all  measurements,  if  possible,  be  made. 

There  are  four  cases. 

*  If  the  two  omitted  sides  are  parallel  and  equal,  their  bearings  cannot 
be  supplied ;  or  if  they  are  parallel  and  of  equal  or  unequal  lengths,  their 
distances  cannot  be  computed. 


SUPPLYING   OMISSIONS. 


167 


CASE  I. 

210.  Given  the  bearings  and  distances  of  all  the  sides  of  a 
tract  of  land  except  the  bearing  and  distance  of  one  side,  to  deter- 
mine these. 

Find  the  latitudes  and  departures  of  the  given  sides.  The 
difference  of  the  northings  and  southings  will  show  the  latitude 
of  the  line  omitted,  and  the  difference  of  the  eastings  and  west- 
ings its  departure.  Then 

Length  of  line  =  Vlat.2  +  dep.2 
Tan  angle  of  bearing  of  line  = 

The  cardinal  points  between  which  the  line  runs  are  indicated 
by  the  deficiency  in  the  latitude  and  departure  columns. 

EXAMPLES. 

1.   Given          (1)  N.  24|°  E.    23.75  chains; 

(2)  S.  81£°  E.    11.70      " 

(3)  S.       1°E.    12.64      " 

(4)  S.  11|°  W.  14.50      " 

To  find  the  length  and  bearing  of  a.  line  connecting  the  ex 
tremity  of  the  fourth  side  with  the  first  corner. 


STA- 
TIONS. 

BEARINGS. 

DISTS. 

N. 

S. 

E. 

W. 

1 

N.  24}°  E. 

23.75 

21.61 

9.85 

.  .  . 

2 

S.  811°  E. 

11.70 

1.78 

11.66 

3 

S.  1°      E. 

12.64 

.  .  . 

12.64 

.22 

4 

S.  11.1°  W. 

14.50 

14.21 

2.89 

21.61 

28.63 

21.63 

2.89 

21.61 

2.89 

7.02  N.    18.74  W. 

Length  of  line  =  V  (7. 02)-'  +  (18. 74)2  =  20.01  chains. 


168 


PLANE   SURVEYING. 


Tan  bearing       = 
Bearing 


18.74 

7.02* 
=  N.  69°  28'  W. 


2.  Given  the  bearings  and  distances  of  the  sides  of  a  tract  of 
land  as  follows  ;  it  is  required  to  find  the  length  and  bearing  of 
the  fourth  side.* 

(1)  N.  llfE.    12.69  chains; 

(2)  S.  87f  W.     8.50      " 

(3)  N.  85£°  W.   11.70      " 
(5)  S.  821°  E.    10.53      " 

The  foregoing  case  may  be  employed  to  overcome  an  obstacle 
in  a  line,  as  LN.  Thus,  surveying  LOPN, 
there  will  be  given  all  the  sides  except  LN, 
which  can  be  determined  as  above.  If  it 
is  desired  to  straighten  an  old  road,  the 
length  and  direction  of  the  new  road  may 
be  computed  from  the  distances  and  deflec- 
tions, or  bearings  of  the  old. 

For  example,  let  ABODE  be  a  crooked 
road  which  it  is  desired  to  replace  by  a  straight  one,  AE.     The 
bearings  and  distances  being  as  follows,  the  length 
and  bearing  of  AE  are  required. 


AB,  North, 
J8C,  N.  20°  E. 
CD,  N.  35°  E. 
DE,  N.  10°  W. 


12.70  chains; 
13.25      " 
12.75      " 
16.90      " 


Ans.    N.  9°  41'  E.  52.98  chains. 

EXAMPLE  2.  Given  the  following  as  the  bearings 
and  distances  of  a  road,  it  is  desired  to  straighten,  to 
find  the  length  and  bearing  of  the  new  road. 


*  In  practice,  the  result  should  be  checked  by  making  a  plot  of  the  field. 


SUPPLYING  OMISSIONS.  169 

(1)  N.  12°    W.  13. 10  chains; 

(2)  N.    8°     E.  1G.20      " 

(3)  N.    21°  W.  14.40      " 

(4)  N.  40J°   E.  15.08      " 

(5)  N.  601°  W.  16.12      " 

EXAMPLE  3.  In  last  figure  but  one,  suppose  LO  bears  N. 
44°  20'  W.,  distance  3.95  chains.  Deflection  at  0  from  OL 
30°,  and  OP=  6.90  chains.  Deflection  at  P  from  OL  100°,  and 
PN=  5.40  chains.  It  is  required  to  find  the  length  and  bear- 
ing of  NL.  Ans.  Bearing  south.  Length,  12.55  chains. 

CASE  II. 

211.  Given  the  bearings  and  distances  of  all  the  sides  of  a 
tract  of  land,  except  the  distances  of  tivo  sides  not  parallel,  to 
determine  these. 

By  Article  168,  change  all  the  bearings  so  that  one  of  the 
sides,  whose  direction  only  is  known,  shall  become  a  meridian. 
Tabulate  the  latitudes  and  departures  corresponding  to  the 
changed  position  of  the  sides.  The  side  made  meridian  will 
have  110  departure,  and  the  difference  of  the  eastings  and  west- 
ings, therefore,  will  be  the  departure  of  the  other  unknown  side. 
Now  with  this  departure  and  the  changed  bearing  the  distance 
and  difference  of  latitude  of  this  side  may  be  found,  and  should 
be  inserted  in  their  proper  places  in  the  table.  Then  the  differ- 
ence between  the  northings  and  southings  will  be  the  latitude, 
or  length  of  the  side  made  a  meridian.* 

212.  Otherwise.     If  the  deficient  sides  adjoin. 

If  a  linef  be  drawn  connecting  L  and  N,  a  figure,  LNOPQ, 
will  be  shown,  in  which  all  the  sides  are  given  except  LN: 
the  bearing  and  distance  of  this  side  may,  therefore,  be  calcu- 
lated by  the  preceding  case.  This  line  and  the  two  sides,  LM 


*It  is  immaterial  whether  or  not  the  deficient  sides  adjoin, 
t  Called  a  closing  line  since  it  closes  the  survey  LQPON. 


170  PLANE   SURVEYING. 

and  MN,  whose  bearings  only  are  given,  will  form  a  triangle, 
in  which  will  be  known  one  side  and  all  the  angles,  whence 
the  unknown  distances,  LM  and  JOT",  may  be  computed. 


213.    If  the  sides  do  not  adjoin. 

In  the  figure  suppose  that  the  distances  LM  and  PO  are 
wanting.  Draw  Ln  and  no  parallel  and  equal  respectively 
to  MN  and  NO.  Then  by  joining  oP,  a  closed  figure  will  be 
formed,  all  the  bearings  and  distances  of  which  are  known 
except  the  bearing  and  distance  of  the  closing  line,  Po,  and 
these  may  be  found  by  Case  I.  Po  thus  determined,  there  will 
be  known  in  the  triangle  PoO  all  the  angles  and  one  side,  to 
find  PO,  and  00,  which  is  equal  to  LM. 

EXAMPLES. 

1.  Given  the  following  bearings  and  distances  of  the  sides  of 
a  tract  of  land,  to  find  the  length  of  the  3d  and  6th  sides. 
(See  last  figure.) 

(1)  N.    61°  W.    9.38  chains; 

(2)  N.  65J°  E.    8.25      " 

(3)  S.   39°     E.    Unknown; 

(4)  S.     2°    W.    4.45  chains ; 

(5)  S.   46°    W.    5.00      « 

(6)  N.  88°    W.    Unknown. 


SUPPLYING   OMISSIONS. 


171 


STA- 
TIONS. 

BEARINGS. 

DlSTS. 

N. 

8. 

E. 

W. 

1 

N.    6i°  W. 

9.38 

9.32 

.  .  . 

1.06 

2 

N.  65£°  E. 

8.25 

3.42 

7.51 

3 

S.  39°    E. 

4 

S.    2°    W. 

4.45 

4.45 

0.16 

5 

S.  4GD    W. 

5.00 

3.47 

3.60 

12.74 

7.92 

7.51 

4.82 

7.92 

4.82 

4.82 

2.69 

Tan  bearing  =  M§,  or  Po  bears  S.  29°  10'  W. 


Length  of  Po  =  V(4.82)2+(2.69)2  =  5.52. 

Angle  P  therefore  =  68°  10'. 
Angle  0        "        =  49°. 

=  62°  50'. 


Angle  o 

sin.  49° 
:  sin.  62°  50' 
:         5.52 

:  PO  (3d  side)  =  6.51 


Ar.  co.  =  0.122220 
=  9.949235 
=i  0.741939 


=  0.813394 


sin.  49°  Ar.  co.  =  0.122220 

:  sin.  68°  10'  =9.967674 

;:         5.52  =  0.741939 

side)  =6. 79  =0.831833 


EXAMPLE  2.   Given  the  following  data  to  supply  the  omissions 

(1)  N.    8£°  E.    9.80  chains; 

(2)  N.  81|°  E.    Unknown ; 

(3)  S.   70°     E. 

(4)  S.     5£°  W.    17,70  chains; 

(5)  N.  87°    W.    18. 75  chains,  to  the  beginning. 


172 


PLANE   SURVEYING. 


EXAMPLE  3.  In  the  last  example  insert  the  distances  found, 
and  suppose  the  first  and  fourth  sides  are  wanting  ;  determine 
these  by  either  or  both  methods. 

CASE  III. 

214.  Given  the  bearings  and  distances  of  all  the  sides  of  o- 
tract  of  land,  except  the  bearings  of  two  sides,  to  determine  these. 
Tabulate  the  latitudes  and  departures  of  the  sides  completely 
given  ;  obtain  the  difference  of  the  northings  and  southings,  and 
of  the  eastings  and  westings.  These  differences  will  be  the 
latitude  and  departure  of  a  closing  line. 

The  bearing  and  distance  of  the  closing  line  may  hence  be 
computed  ;  then  in  the  triangle  formed  by  this  line  and  the  two 
sides  whose  distances  are  given,  determine  the  angles  ;  and 
thence,  with  a  proper  application  of  them  to  the  bearing  of  the 
closing  line,  the  wanting  bearings  may  be  found. 

In  the  figure  let  PQONML  represent  a  tract  of  land  in  which 
all  the  bearings  and  distances  are 
known  except  the  bearings  of  QO 
and  NM. 

Drawing  nN  parallel  and  equal  to 
QO,  and  joining  Qn  and  nM,  a 
closed  figure,  PQnMLP,  will  be 
formed,  in  which  the  bearing  and 
distance  of  nM,  the  closing  line, 
may  be  calculated  by  Case  I.  Then 
in  the  triangle  MnN,  having  all  the 
sides,  the  angles  are  readily  found,  and  by  proper  applica- 
tion of  these  with  the  bearing  of  Mn  the  bearings  of  NM,  and 
nN=  QO  may  be  obtained. 

NOTE.  —  If  the  sides  whose  bearings  are  required  adjoin,  the  reasoning 
is  evident.  If  they  do  not  adjoin,  a  transposition  of  some  of  the  sides  may 
be  made,  as  in  the  preceding  case,  without  changing  the  direction  or  length 
of  any  of  them,  making  the  unknown  sides  adjoin,  and  with  the  closing 
line  form  the  triangle  referred  to  in  the  last  paragraph.  The  rule  is, 
therefore,  applicable  to  either. 


SUPPLYING   OMISSIONS. 


173 


EXAMPLES. 

1.    Given  the  following  data  of  a  survey,  to  supply  the  omis- 
sions.    Referring  to  the  last  figure  : 


e  bearing  of 

PQ, 

N. 

3° 

E. 

dist. 

4 

.57 

chains. 

(4 

u 

QO, 

tt 

6.25 

ii 

II 

II 

ON, 

S. 

23| 

°E. 

!( 

5 

.50 

(t 

(( 

11 

NM, 

(( 

4 

.88 

it 

!< 

II 

ML, 

N. 

87° 

W. 

II 

•2 

.97 

u 

(( 

II 

LP, 

N. 

43° 

W. 

U 

3 

.:},'} 

K 

STA- 
TIONS. 

LINES. 

BEARINGS. 

DlSTS. 

N. 

8. 

E. 

W. 

P 

PQ 

N.3°      E. 

4.57 

4.56 

.  .  . 

0.24 

.  .  . 

Q 

0,0 

6.25 

0 

ON 

S.  23£°  E. 

5.50 

.  .  . 

5.05 

2.19 

.  .  . 

N 

NM 

433 

M 

ML 

N.  87°    W. 

2.97 

0.15 

... 

... 

2.97 

L 

LP 

N.  43°    W. 

3.33 

2.43 

2.28 

7.14 

5.05 

2.43 

5.25 

5.05 

2.43 

Deficiency,    2.09  S.                Deficiency,   2.82  E. 

Tan  of  bearing  of 


^=-,  and  bearing  =  S.  53°  28'  E. 


Dist.  nM=  V(2.09)2  +  (2.82)2  =  3.51. 
To  find  the  angle  of  nMN: 

log  4.33     Ar.  co.  =        9.363512 
log  3. 51  "      =        9.454693 

log  7.045  =        0.847881 

log    .795  =        1.900367 

2)19.566453 
log  cosine  £  nMN=       9.783226 

and  £  <  =    52°  37' 
2_ 

105°  14' 


174  PLANE   SURVEYING. 

Now,  since  Mn  bears  N.  53°  28'  W.,  and  the  angle 
=  105°  14',  the  line  MN  is  in  the  northeast  quadrant,  and 
makes  an  angle  with  the  meridian  =  105°  14'  —  53°  28'  =  51°  46', 
or  its  bearing  is  N.  51°  46'  E. ;  and  hence,  reading  in  the  order 
the  measurements  were  made,  the  bearing  of  NM=  S.  51°  46'  W. 

To  find  the  angle  nNM,  and  thence  the  bearing  of  QOs 

6.25  Ar.  co.  =  9.204120 

:3.51  =0.545307 

: :  sin  105°  14'  =  9.984466 


:  sin  32°  48'  «nNM)  =  9.733893 

Bearing  of  NM  =  S.  51°  46'  W. 

<  nNM  32°  48'  on  west  side,  add       32°  48' 

Bearing  of  Nn  =  OQ  =  S.  84°  34'  W., 

or  bearing  of  QO  =  N.  84°  34'  E. 

2.   Supply  the  omissions  from  the  following  data : 

(1)  N.  34°    W.     13. 00  chains; 

(2)  S.  41|°  W.     12.90      " 

(3)  S.  50°     E.       8.20      " 

(4)  2.56      " 

(5)  6.90      " 

(6)  N.  28°     E.       9.95      «« 

CASE  IV. 

215.  Given  the  bearings  and  distances  of  all  the  sides  of  a 
tract  of  land  except  two,  one  of  tvhich  has  only  its  bearing  given, 
and  the  other  the  distance,  to  supply  the  omissions. 

Make  a  meridian  the  side  whose  bearing  only  is  given.  Tab- 
ulate the  latitudes  and  departures  corresponding  to  the  changed 
position  of  the  survey.  The  side  made  meridian  will  have  no 
departure,  and  the  difference  of  the  eastings  and  westings, 
therefore,  will  be  the  departure  of  the  side  whose  bearing  is 
unknown.  With  the  given  distance  and  this  departure  the 


SUPPLYING  OMISSIONS. 


175 


changed  bearing  and  difference  of  latitude  of  this  side  may  be 
found,  and  should  be  inserted  in  their  proper  places  in  the 
table.  Then  the  difference  of  the  northings  and  southings  will 
be  the  latitude,  or  length,  of  the  side  made  a  meridian. 

216.  Otherwise.      When  the  deficient  sides  adjoin. 

Let  the  bearing  of  MN  and  the  distance  LM  be  wanting. 
Calculate  by  Case  I.  the  direction 
and  length  of  the  closing  line  LN. 
A  triangle,  LMN,  may  then  be  formed 
in  which  will  be,given  the  lengths  of  Q 
LN  and  MN,  and  the  angle  NLM. 
The  distance  LM  and  the  angle  N 
may  therefore  be  computed,  and  the 
angle  N  thus  found  properly  applied 
to  the  bearing  of  the  closing  line  will 
give  the  bearing  of  MN.  L  M 

217.  When  the  deficient  sides  do  not  adjoin. 

Referring  to  the  same  figure  as  before,  suppose  the  bearing 
of  LM  and  the  distance  OP  wanting.  Transpose  the  sides 
as  there  shown,  and  calculate,  as  in  Case  I.,  the  direction  and 
length  of  the  closing  line  Po.  Then,  as  in  the  preceding  article, 
there  will  be  given  a  triangle,  OPo,  in  which  are  known  two 
sides  Po  and  Oo,  and  the  angle  P,  whence  the  bearing  of  Oo, 
or  LM,  and  the  distance  PO,  may  be  determined. 


EXAMPLES. 
1.     Given  the  following  notes,  to  supply  the  omissions. 


QP. 
PO. 
ON. 


N.  10° 
S.  88£ 
S.  16 


E. 
E. 
E. 


13.71  chains; 
18.75      " 
16.50      «« 
13.00      «* 


NM.   S.  W. 

ML.    N.  80°    W. ; 

LQ.     N.  36°    W.     10.00  chains. 


176 


PLANE   SURVEYING. 


STA- 
TIONS. 

LINES. 

BEARINGS. 

DIXTS. 

N. 

8. 

E. 

W. 

Q 

QP 

N.  10°    E. 

13.71 

13.50 

2.38 

P 

PO 

S.  88£°  E. 

18.75 

.  .  . 

0.49 

18.74 

.  .  . 

0 

ON 

S.  16J°  E. 

16.50 

.  .  . 

16.82 

4.68 

N 

NM 

13.00 

M 

ML 

N  80°    W 

L 

LQ 

N.  36°    W. 

10.00 

8.09 

5.88 

21.59 

16.31 

25.80 

5.88 

16.31 

5.88 

Deficiency,  5.28  S.           Def.,  19.92  W. 

Tan  of  bearing  of  closing  line  = 


19.92 


5.28' 


or  bearing  of  NL,  S  75°  09'  W. 


Length  of  NL  =  V(5.28)2+(19.92)2  =  20.61, 

and  angle  MLN =  24°  51'. 
To  find  angle  LMNi 

13.00    (NM)     Ar.  co.  =  8.886057 

:  sin.  24°  51'  «L)  =  9.623502 

::    20.61  (LN)        =  1.314078 

:  sin.  41°  47'  =  9.823637 

Angle  LMN=  180°  -  41°  47'=  138°  13'  (see  note) 
Angle  LMN=  180°  -  (138°  13'+24°  51')  =  16°  56'. 


NOTE.  —  When  the  side  MN,  whose  length  only  is  given,  is  longer  than 
the  closing  line  LN,  the  angle  M  must  be  acute ;  if  shorter,  the  angle  M 
may  be  acute  or  obtuse,  depending  upon  the  length  of  the  side  LM,  the 
bearing  of  which  only  is  known.  Hence,  when  this  last  relation  obtains,  it 
is  necessary,  in  the  application  of  this  case,  to  remove  ambiguity,  that  enough 
be  known  concerning  the  length  of  the  side,  whose  bearing  only  is  given, 
to  indicate  whether  the  angle  M  is  greater  or  less  than  a  right  angle. 

In  the  example,  LM  is  known  to  be  shorter  than  NXf,  and  hence  angle 
M  is  obtuse.  The  ambiguity  is  not  removed  by  employing  the  method 
given  in  Article  215. 


PLOTTING    A   COMPASS   OR   TRANSIT   SURVEY.         177 

The  bearing  of  NM,  S.  75°  09'  W— 16°  56'=  S.  58°  13'  W. 

To  find  the  length  of  LM : 

sin.  24°  51'  Ar.  co.  =  0.376498 

:  sin.  16°  56'  =  9.464279 

::          13.00  =1.113943 


:  9.01  (LM)  =  0.954720 

The  student  may  verify  by  the  method  in  Article  215. 

EXAMPLE.  As  an  exercise,  from  any  of  the  preceding 
problems  strike  out  from  two  sides  that  do  not  adjoin  the  bear- 
ing of  one  and  the  distance  of  another,  and  compute  them. 


SECTION  VIII. 

PLOTTING    A    COMPASS    OR    TRANSIT    SURVEY. 

218.  In  addition  to  the  drawing-instruments  explained  in 
chain  surveying,  the  draughtsman  will  now  find  very  convenient 
an  instrument  for  measuring  angles,  or, 

A  Protractor.  It  is  made  of  metal*  or  paper,  usually  in  the 
form  of  a  semi-circle,  the  arc  of  which  is  divided  into  180  equal 
parts,  or  degrees,  subdivided  and  numbered  both  ways. 

To  draw  a  line  making  a  given  angle  with  another  at  a  certain 
point.  Bring  the  diameter  of  the  protractor  to  coincide  with 
the  given  line,  its  centre  with  the  point,  and  the  arch  lying  in 
the  direction  of  the  desired  line  ;  then  with  a  sharp  pencil  or  fine 
needle  prick  off  the  required  number  of  degrees ;  joining  the 
point  thus  fixed  and  the  given  point  completes  the  problem. 

Some  plain  scales  are  graduated  to  degrees  on  three  edges  so 

*  For  more  accurate  work  there  is  attached  a  movable  arm  or  ruler, 
extending  beyond  the  circumference  and  carrying  a  vernier. 

12-inch  protractors, —  complete  circle, —  made  of  heavy  paper,  on  which 
are  printed  the  divisions  to  quarter-degrees,  are  quite  reliable. 


178  PLANE   SURVEYING. 

as  to  be  used  like  a  protractor,  but  are  objectionable  on  account 
of  the  obliquity  of  the  divisions  and  their  varying  lengths. 


219.  Illustration.  To  plot  a  survey  the  record  of  which  is 
as  follows : 

(1)  N.  llf°  E.     13.19  chains; 

(2)  S.  87°    W.       8.50      " 

(3)  S.  2C4°  W.     11.75      " 

(4)  S.  82°     E.     10.03      " 

With  a  Protractor.  First  Method.  Represent  the  meridian 
by  drawing  on  the  paper  a  line  so  situated  that  there  will  be 
sufficient  room  on  either  or  both  sides  of  it,  as  the  case  may  be, 
to  complete  the  drawing.  Fix  upon  a  point  in  this  line  to 
indicate  a  corner  of  the  tract,  usually  "  the  place  of  beginning." 
In  this  particular  example  the  first  corner  is  the  easterly  boun- 
dary, and  as  it  runs  northerly,  we  will  draw  our  meridian  near 
the  lower  right-hand  side  of  the  paper,  as  at  A.  Prick  off  the 
angle  11|°  from  the  north  end  of  the  protractor-arch  to  the 
right,  and  draw  the  line  13.19  chains  (AE)  to  any  convenient 
scale,  say  2  chains  to  an  inch,  or  6.6  inches.  Pass  another 
meridian  N'S'  through  B ;  and  since  the  bearing  is  south- 
westerly, we  prick  off  the  degrees,  87  from  the  south  point,  and 
draw  the  line  8.50  (BC)  to  the  same  scale.  In  a  similar  man- 


PLOTTING   A   COMPASS   OR   TRANSIT   SURVEY.        179 


ner  draw  the  line  CD,  and  finally  DA,  which  should  end  at  A. 
Jf  it  does  not  end  precisely  at  A,  an  error  in  plotting,  or 
inaccuracy  in  the  survey,  would  thereby  be  indicated. 

An  error  in  plotting  a  line  by  this  method  would  affect  the 
position,  but  not  the  direction  of  the  following  lines. 


220.  Another  Method.  By  laying  off  the  angles  from  one 
point,  or  from  one  position  of  a  protractor  having  a  complete 
circle.  With  the  protractor  at  any  convenient  point,  P,  in  the 
meridian  NS,  prick  off  the  degrees  shown  by  the  bearings, 
and  indicate  each,  and  the  side  to  which  it  belongs,  as  per 
figure.  Then,  by  instruments  used  for  drawing  parallel  lines, 
transfer  them  to  their  proper  places,  and  make  the  lengths  cor- 
respond to  the  scale  adopted.  The  point  P,  from  which  all  the 
angles  were  set  off,  may  or  may  not  be  one  of  the  corners  of 
the  field.  The  figure  shows  that  it  saves  one  transfer  if  so 
taken. 


180 


PLANE   SURVEYING. 


EXAMPLES. 

1.  Plot    a    triangle,    given    tw<5    sides    and    the    included 
angle. 

2.  Given   two   angles   and  the   included    side,   to  plot   the 
triangle. 

3.  Given    three  sides  and  two  included   angles,  to  plot  a 
trapezium. 

QUERY.  Can  a  trapezium  be  plotted  when  there  are  given  all 
the  sides  and  one  angle? 

221.  By  Latitudes  and  Departures.  The  survey  being  bal- 
anced, this  is  the  most  accurate  method,  and  is  equally  appli- 
cable to  a  compass  or  transit  survey. 

Taking  the  record  of  the  survey  in  Article  208,  and,  using 
the  corrected  latitudes  and  departures,  let  us  make  a  plot 
of  it. 

,q  Draw  through   the   first  sta- 

tion* (M)  a  meridian,  and  find, 
by  algebraic  additions,  from  the 
columns  of  corrected  latitudes 
and  departures,  the  distance 
each  corner  is  north  or  south 
from  this  station,  called  total 
latitude,  and  east  or  west  from 
the  meridian  called  total  depar- 
ture. These  distances  may  be 
ascertained  mentally  as  we  pro- 
ceed with  the  drawing,  but  to 
avoid  error  it  is  best  to  tabulate 
them,  using  three  columns,  as 
follows  :  +  indicates  distance 
north  or  east,  and  — ,  south  or 
west,  from  the  references. 


*  Any  station  will  answer,  but  the  one  through  which  the  meridian  is 
supposed  to  pass  in  calculating  the  area  is  preferable. 


PLOTTING   A    COMPASS   OK   TRANSIT   SURVEY. 


181 


STATIONS. 

TOTAL  LATITUDES 

FROM 

STATION  M. 

TOTAL  DEPARTURES 

FROM 

MERIDIAN  NS. 

1 

0 

0 

2 

-12.27 

+    4.68 

3 

-   3.21 

+  14.84 

4 

+    0.20 

+  11.58 

5 

+   3.54 

+    8.23 

6 

+    1.97 

+    6.25 

1 

0 

0 

The  total  latitude  of  the  last  station  is  the  latitude  of  the  last 
line  with  its  sign  changed.  The  same  is  true  regarding  the 
total  departure  of  last  station.  A  check  is  thus  had  on  the 
work. 

From  M  lay  off  on  the  meridian  negatively,  or  to  the  south, 
12.27  chains  according  to  the  scale  adopted,  to  A  ;  from  A  set 
off  perpendicularly  to  the  east,  with  the  same  scale,  4.68  chains, 
to  0 ;  connect  M  and  0,  showing  the  first  line.  Set  off  from 
3/,  again  to  the  south,  3.21  chains,  to  B;  thence  perpendicu- 
larly to  the  right,  or  east,  14.68  chains,  to  P. 

OP  represents  the  second  line  of  the  survey.  Next  lay  off 
20  links  to  the  north  from  3f,  thence  11.58  chains  to  the  east, 
and  join  PQ  for  the  third  line,  and  so  on,  the  last  line,  SM, 
requiring  a  distance  laid  off  on  the  meridian  north  =  1.97  ;  and 
a  perpendicular  thereto,  =  6.25  east,  when  drawn  closes  the  sur- 
vey, thus  proving  the  correctness  of  the  work. 

A  variation  of  the  method  just  given  is  to  draw  two  lines,  one 
representing  the  meridian,  the  other  an  east  and  west  line.  On 
the  first  lay  off,  as  before,  the  latitudes  of  the  sides,  and  on 
the  second  the  corresponding  departures;  then,  by  means  of 
dividers,  obtain  the  intersection  of  co-ordinates,  and  joining 
these  points  shows  the  plot. 

For  plots  of  ordinary  farm  surveys  the  method  given  above, 
being  equally  accurate  and  more  expeditious,  is  recommended ; 


182 


PLANE   SURVEYING. 


for  plots  of  extraordinary  size,  extending  over  a  large  drawing- 
board  or  made  on  a  large  table,  the  variation  noted  should  be 
adopted. 

222.  Using  Cross-Section  Paper*  and  the  latitudes  and  de- 
partures, a  tolerably  accurate  plot  may  be  made  with  great 
facility.  The  vertical  and  horizontal  lines  of  the  paper  may 


represent  respectively  meridians,  and  east  and  west  lines.  As- 
sume any  convenient  point  0  as  the  beginning  of  the  survey, 
and  suppose  the  latitude  of  the  first  line  =  4. 00  chains  N.,  the 
departure  =  6.00  chains  E.  Count  from  0  northward  four 
spaces,  thence  eastward  six  spaces,  to  P ;  join  OP,  thus  deline- 
ating the  first  side.  Suppose  the  latitude  and  departure  of  the 


*  Note-books  may  be  procured   having  the   alternate   pages  ruled   in 
small  squares,  like  cross-section  paper. 


DETERMINING   AREAS.  183 

second  side  =  respectively  3.50  chains  N.  and  2.25  chains  E. ; 
count  off,  as  before  (estimating  the  fractions  of  chain),  three 
and  a  half  spaces  north  and  two  and  a  quarter  east ;  connect 
the  points  P  and  Q  for  the  second  side,  and  so  on  to  the  place 
of  beginning. 


SECTION  IX. 
ON  DETERMINING  AKEAS. 

A.    PARTICULAR  FORMS  AND  CASES. 
TRIANGLES.* 

223.   First  Method.     Measure  two  sides,  as  m  and  n,  and  the 
included  angle  0.     Then  the 


224.  Second  Method.      Measure   two  m 
angles,  as  0  and  N,  and  the  included  side  m.     Then 

A_  m2sin  JV'sin  O 
~  2s\n(N+0)' 

PARALLELOGRAMS. 

225.  Measure  two  adjacent  sides,  m  and  n,  and  their  in- 
cluded angle,  P.     Then  h  denoting 

the  altitude, 

A.  =  mh  =  m  x  n  sin  P. 


*  For  other  methods  than  those  found  in  this  section  of  surveying  tri- 
angles, quadrilaterals,  and  other  polygons,  see  Chain  Surveying,  Articles 
03  to  70. 


184  PLANE   SURVEYING. 


EXAMPLES. 

1.  Two  sides  of  a  triangle  measure  756  feet  and  1024  feet, 
and  their  included  angle  42°  45' ;  determine  the  area  in  acres. 

2.  Two  angles  of  a  triangle  are  59°  29'  and  65°  18',  and  their 
included  side  932  feet.    How  many  acres  does  it  contain?    Plot. 

3.  Two  sides  of  a  triangle  measure  15.24  chains  and  13.18 
chains,  and  the  angle  opposite  the  first  54°  25'.     Find  the  area. 

4.  Two  adjacent  sides  of  a  parallelogram  are  856  feet  and 
1252  feet,  and  their  included  angle  75°  48'.     Compute  the  area. 


TRAPEZOIDS. 

226.   Measure  three  sides,  say  PM,  MN,  and  NO,  and  one 
of   the  included   angles,  as  N.      From 

?        the  data  thus  obtained  compute  the  al- 

j\       titude,  OL  =  PK,  and  the  parallel  side, 
j  \      PO.     Then 
M    K  L  N  A  =  MN+  PO  x  pK 

Or,  instead  of  measuring  the  inclined  sides,  if  it  is  equally 
convenient  measure  the  parallel  sides,  and  one  of  the  other 
sides  and  an  angle  as  before ;  then 


TRAPEZIUMS. 

227.  Measure  all  the  sides  and  one  angle.  With  the  data 
calculate  the  length  of  a  diagonal  dividing  the  tract  into  two 
triangles,  in  one  of  which  two  sides  and  the  included  angle  will 
be  given,  and  in  the  other  three  sides,  whence  the  area  may  be 
found. 


DETERMINING   AREAS.  185 

228.  Or,  measure  three  sides,  PM,  MN,  and  ON,  and  the 
included  angles  N  and  PMN.      Draw  Pr 

MO,  calculate  the  area  of  the  triangle 
MNO,  the  diagonal  MO,  and  the  angle 
OWN.  Subtract  OMN  from  PMN; 
then,  having  two  sides  and  the  included 
angle  in  the  triangle  PMO,  its  area  may 
be  computed,  which  added  to  the  area  of  MNO  gives  the  re- 
quired content. 

229.  Otherwise.  Measure  two  opposite  sides,  as  OL  and  MN, 
and  three  angles,,  as  0,  L,  and  M. 

Conceive  the  sides  ON  and  LM  to 
be  prolonged  to  meet  in  some  point, 
P.  From  the  data  calculate  the 
areas  of  the  triangles  POL  and 
PMN.  The  difference  will  give  the  area  sought.* 

EXAMPLES. 

1.  Given  in  a  trapezoid  (see  figure,  Article  226)  PM  =  33 
rods,  MN=  68  rods,  NO  =  30  rods,  and  the  angle  N=  70°  ;  to 
find  the  area,  and  make  a  plot. 

2.  Given  in  a  trapezium  PMNO   (see  figure,  Article  228) 
PM=  7  chains,  MN=  7.50  chains,  NO  =  6  chains,  the  angle 
2V=120°,  and  M=  108° ;  to  find  the  area,  and  make  a  plot. 

3.  Given  in  a  trapezium  LMNO  (see  last  figure)  L0  =  8 
chains,  MN=  5  chains,  and  the  angles  L,  M,  and  N  respectively 
87°,  70°,  and  80° ;  to  find  the  area,  and  make  a  plot. 

4.  Given  in  a  trapezium  the  angle  Ma  right  angle,  the  sides 
MN,  NO,  OP,  and  PM  respectively  20,  12,  30,  and  15  rods; 
also  a  perpendicular  to  MN  from  N  extending  to  P0=10 
rods  ;  to  find  the  area. 

QUERY.  Could  the  area  be  found  without  NO? 

*  If  practicable,  observe  all  the  angles,  and  thereby  obtain  a  check  on 
the  measurements. 


186 


PLANE   SURVEYING. 


POLYGONS- 

230.  To   find   the   area   of    an   irregular    pentagonal    field 
LMNOP,  when  all  the  corners  can   be  seen  from  one  corner, 

as  0.  Measure  the  sides  ON,  OP,  the  diagonals 
OL,  OM,  and  the  three  angles  at  0.  Then  two 
sides  and  the  included  angle  of  each  triangle 
thus  formed  will  be  given,  whence  their  areas 
may  be  calculated  and,  by  addition,  the  area  of 
the  required  polygon  may  be  obtained.  In  like 
manner,  a  survey  of  any  small  irregular  polygonal 
lot,  in  which  all  the  corners  are  visible  from  one 
corner,  may  be  effected.  If  there  are  n  sides, 
measure  from  one  corner  two  sides  and  n  —  3  diagonals,  observ- 
ing from  the  same  point  the  n  —  2  angles  which  are  formed  by 
these  diagonals  and  the  two  sides.  Then,  as  above,  the  tract  will 
be  divided  into  n  —  2  triangles,  the  area  of  each  may  be  calculated, 
and  the  sum  of  these  areas  taken  for  the  area  of  the  polygon. 

231.  Or,   measure  from  some  point  within  or  without  the 
field  radial  lines  to  all  the  corners,  and  observe  at  the  same 
point  the  angle  which  these  lines  make  with  each  other. 

There  will  thus  be  given  two  sides  and  the  included  angle  of 
a  series  of  triangles,  whence  the  bounding  lines  and  area  may 
be  computed. 

232.  Othei-wise.     Measure  a  base  line  within  or  without  the 
tract,  or  use  a  portion  or  all  of  one  side  as  a  base  line,  and 

observe  from  each  extremity  of  this 
line  the  angles  formed  by  it  and  a 
visual  line  through  each  corner  of 
the  tract.  There  will  thus  be  known 
two  angles  and  the  included  side  of 
a  series  of  triangles,  whence  the 
bounding  lines  and  area  may  be 
calculated. 

The  marginal  figure  represents  the 


DETERMINING   AREAS. 


187 


case  where  the  base  line  BL  is  taken  outside  the  tract.  It  will 
be  noticed  that  it  is  possible  by  this  method  to  survey  a  farm 
without  entering  upon  it. 


B.     GENERAL  METHOD. 

233.  The  methods  given  in  the  last  three  articles  are,  however, 
quite  limited  in  their  application,  since  it  rarely  happens  in  a 
tract  of  considerable  magnitude  that  all  the  corners  are  visible 
from  any  one  corner,  or  from  any  point  within  or  without  the 
field. 

The  following  niethod  of  determining  the  area  by  means  of 
latitudes  and  departures  is  applicable  to  all  right-lined  figures, 
and  is  the  most  general  and  accurate. 

Let  POLM  represent  a  tract  of  land,  the  area  of  which  is 
desired.    Measure  all  the  sides  and  angles,  interior  or  deflection, 
with  a  single  bearing,  if  the  transit 
is  used,  or  take  all  the  bearings  with 
a  compass.     Distribute  the  angular 
error,  if  any  made  by  transit  (see 
note,  Article  207).    Obtain  the  lati- 
tudes and  departures,  and   balance 
the  survey. 

Let  NS  represent  a  meridian  pass- 
ing through  P,  the  most  westerly 
station  of  the  tract,  and  .B&,  C'c,  and 
Dd  meridian  distances.  Now,  if 
perpendiculars  be  dropped  from  the 
angles  0,  -L,  and  M to  the  meridian, 
it  will  readily  appear  that  the  area  of 
POLMP=  area  oOLMmo  minus  the 
sum  of  the  areas  of  the  triangles  PoO 
and  PmM,  or  POLMP  =  trape- 
zoids  (OoLl  +  LlmM)  —  triangles 
(PoO  +  PmM} 

=  Ccx  OQ  +  Dd  x  LR  —  Bb  x  Po  —Ee  x  Pm. 


N 


Ifl 


188  PLANE   SURVEYING. 

The  computation,  then,  involves  the  latitudes  and  depart- 
ures, and  meridian  distances  ;  the  former  having  been  already 
explained,  we  shall  now  indicate  how  the  latter  may  be  obtained, 
or  rather  how  the  double  meridian  distances  are  found,  since  in 
order  to  lessen  fractions  the  double  lengths  are  used. 

The  double  meridian  distance,  or  D.M.D.,  of  the  side 
PO  =  2Bb  =  Oo,  its  departure. 

The  D.M.D.  of  OL  =  2  Cc  =  Oo  +  Ll=  Oo  +  QI+QL 
=  2Bb+Oo+QL. 

The  D.M.D.  of  LM=  2  Dd=  2  Cc  +  QL  -  MR. 

The  D.M.D.  of  MP  =  2Ee  =  2Dd-  MR  -  Mm 

=  Mm  =  its  departure.  . 

It  is  evident  that,  in  a  corresponding  manner,  the  double 
meridian 'distances  of  the  bounding  lines  of  a  tract  may  be 
found,  no  matter  what  the  number  of  sides  or  magnitude  of  the 
angles.  Hence,  considering  east  departures  plus  (  +  ),  and 
west  departures  minus  (  — ),  the  above  deductions  may  be 
expressed  in 

A  GENERAL  RULE  FOR  OBTAINING  DOUBLE  MERIDIAN 
DISTANCES. 

The  double  meridian  distance  of  the  first  side  is  equal  to  its 
departure. 

The  double  meridian  distance  of  the  second  side  is  equal  to  the 
double  meridian  distance  of  the  first  side,  plus  its  departure,  plus 
the  departure  of  the  second  side. 

The  double  meridian  distance  of  any  side  is  equal  to  the  double 
meridian  distance  of  the  preceding  side,  plus  its  departure,  plus 
the  departure  of  the  side  itself. 

The  double  meridian  distance  of  the  last  side  deduced  by  the 


DETERMINING    AREAS. 


189 


foregoing  rule  should  equal  its  departure,  and  ivill  serve  as  a 
check  on  this  part  of  the  work.* 

234.   Continuing   now   the   work   of    computing   areas    and 
referring  to  the  last  figure,  we  may  form  the  following  table  : 


STATIONS. 

LINES. 

N.  LAT. 

S.  LAT. 

E.  DEP. 

w. 

DEP. 

D.M.D. 

NORTH 
DOUBLE 
AREAS. 

SOUTH 
DOUBLE 
AREAS. 

1 

PO 

Po 

Oo 

. 

2  Bb 

2  Bb  X  Po 

2 

OL 

.  . 

OQ 

QL 

2  Cc 

.... 

2CcxOQ 

3 

LM 

Lli 

MR 

2  Dd 

2Dclx  LR 

4 

MP 

Pm 

Mm 

2Ee 

2EeXPm 



The  double  meridian  distances  are  placed  in  the  column 
headed  D.M.I).  In  the  column  headed  North  Double  Areas  are 
placed  '2Bb  X  Po  and  2Ee  X  Pm,  the  product  of  the  first  and 
fourth  double  meridian  distances  and  their  corresponding  lati- 
tudes. In  the  south  double  area  column  we  find  2  Cc  x  OQ 
and  2  Dd  X  Lit,  or  the  product  of  the  double  meridian  distances 
of  the  second  and  third  sides,  and  their  respective  latitudes. 
In  other  words,  the  column  in  which  each  of  the  products  of 
double  meridian  distance  and  latitude  is  to  be  placed  is  indi- 
cated bv  the  latitude  employed  in  the  multiplication. 

Now,  twice  the  area  of  the  triangles  POo  and  PMm,  or  the 
subtractive  portion  of  the  figure  oOLMmo,  is  given  in  the  north 
double  area  column,  and  twice  the  area  of  the  trapezoids  OoLl 
and  LI  Mm,  which  include  the  triangles  named,  is  given  in  the 
column  of  south  double  areas.  Half  the  difference,  therefore, 

*  The  position  of  the  meridian  (NS)  may  be  assumed  to  pass  through 
any  other  corner,  or  even  through  a  point  outside  the  survey.  A  slight 
modification  of  the  rule  just  given  would  make  it  applicable  to  any  of 
these  cases.  For  convenience,  it  is  generally  assumed  to  pass  through 
the  most  westerly  station.  When  a  survey  is  made  with  the  transit,  and 
the  area  only  required,  it  is  most  convenient  to  consider  one  of  the  sides 
of  the  tract  the  meridian. 


190  PLANE   SURVEYING. 

between  these  sums  is  the  area  POLMP  required.  The  reason- 
ing'being  general,  and  independent  of  the  number  of  sides,  we 
have  for  finding  the  area  of  any  rectilineal  figure,  the  bearings 
and  distances  of  all  the  sides  being  known,  the  following 

RULE. 

1.  Prepare  a  table  as  exhibited  below. 

2.  Find,  and  place  in  their  proper  columns,  the  latitudes  and 
departures  of  the  several  sides  of  the  tract. 

3.  Balance  the  survey  (if  necessary) . 

4.  Find  the  double  meridian  distances,  with  reference  to  a 
meridian  passing  through  the  most  westerly  *  station,  and  place 
them  in  the  D.M.D.  column. 

5.  Multiply  each  double  meridian  distance  by  its  corresponding 
corrected  latitude,  and  place  the  product  in  the  column  of  double 
areas  indicated  by  the  latitude. 

6.  One-half  the   difference  of  the   sums   of  the  columns   of 
double  areas  will  be  the  required  area. 

Let  us  now  take  the  field  notes  given  in  Article  198,  and  com- 
pute the  area  of  the  tract. 

The  student  will  perceive  that  the  meridian  is  assumed  to 
pass  through  the  most  westerly  station,  that  the  double  meridian 
distances  are  found  as  directed  in  233,  that  each  is  multiplied 
by  its  corresponding  latitude,  and  the  resulting  double  area 
product  placed  in  the  column  of  the  same  name  as  the  latitude. 

Lastly,  the  difference  of  the  two  columns  of  double  areas  is 
taken,  the  remainder  divided  by  two,  giving  the  number  of 
square  chains  in  the  tract,  and  the  result  divided  by  10  =  12,032 
acres,  which  is  the  area  sought. 

On  account  of  the  meridian  passing  through  the  most  west- 
erly station,  and  because  the  field  is  to  the  left,f  the  areas  of 

*  For  convenience  simply,  see  note,  preceding  article. 

t  In  the  last  figure,  if  the  bearings  are  taken  or  recited  in  the  order 
PM,  ML,  LO,  etc.,  the  tract  is  considefed  on  the  left;  if  this  order  is 
reversed,  the  tract  is  on  the  right. 


DETERMINING    AREAS. 


191 


w 

-4 

O5    en    tf»-    co    to    H-> 

STATIONS. 

cc   02    2!    fe!    tej    cr> 

*J       Qt       .is.        rf^       il^.       K5 

fc^     H-*     ^     CO     GO      O 

I  i  |  |  s  s 

^  ^  ^  ^  w  w 

td 
H 
1 

05     fcO     *•     rf^     CO     CO 
S     OS     Cn     05     §     P 

DISTANCES. 

Oi     Oi 

^-q    bo 

oo     en 

co    co    jo    • 

!        .'        en     fcS     CO     ! 

S 

. 

LATITUDES. 

nks.  Error  4  links.  2)240.6401 
10)120.32005 
12.032  acres. 

Cn 
~i 

CO 

,_,    l_l    •       •       •       to 

O     Cn     •        '        '        t? 

&3 

£ 
bo 
t* 

0     H^ 

•_    as 

.         .         .         Oi     *^J 

bq 

DEPABTUBES. 

bo    be 

t*     01 

O5      H-i      CO      CO      • 

^  s  §§  g  :  : 

S 

-a 

h-»      ]          H-"      h->      bS      tsS 

H 

a. 

COBBEC- 

TIONS. 

i*> 

,-  ;    -  ;    MM 

i 

in 

CC 

co    co    co    • 

:  :   ss  fc  §  : 

•^ 

COBBECTED 
LATITUDES. 

ei 

CO 

M 

HJ        h-»        '             '             '             t» 

^s  25  :   :  :  is 

?> 

* 

% 

s  *> 

'.        '.        '.        '.        0     00 

!• 

COBBECTED 
DEPABTUBES. 

is 

o    H->    co    co    • 

g  s  g  s  :  : 

^ 

0      1^      S      05      S      *• 

.if,  &  2  fe  g  § 

o 

k  - 
b 

ii     li 

:   :   g  |  |  : 

Hi 

f 

K  p  :    ;    ;    25 

CO      ^J                _         _          *. 

^      CO      .          .          .          CO 

Cn      OS                                    O5 

Hi 

192 


PLANE   SURVEYING. 


the  trapezoids  are  north,  and  those  of  the  triangles  south.  If 
we  had  assumed  the  meridian  to  pass  through  the  most  easterly 
corner,  the  areas  of  the  trapezoids  then  formed  would  be  south, 
and  those  of  the  triangles  north. 

If  the  bearings  of  the  lines  were  reversed,  or  the  survey  made 
with  the  field  to  the  right,  the  reverse  of  the  preceding  state- 
ment would  be  true. 

In  either  case,  however,  one-half  the  difference  of  the  sums 
of  the  double  areas  will  give  the  contents. 

As  an  exercise  the  student  may  obtain  an  expression  for  the 
area  of  POLMP,  last  figure,  assuming  the  meridian  to  pass 
through  L,  and  passing  i-ound  by  MP,  etc.,  that  is,  keeping  the 
field  to  the  right.  He  may  also,  with  the  meridian  through  P,  and 
keeping  the  field  to  the  left,  obtain  an  expression  for  the  area. 

As  a  further  exercise  he  may  verify  the  result  in  the  last 
example  solved,  taking  the  meridian  through  the  most  easterly 
station. 

Calculate  the  areas  from  the  following  notes ;  also  make  a 
plot  of  each : 

1. 

9°  W. 
31°  E. 
69°  E. 
0^°  E. 
Of  W. 
89°  W. 


(1)  N. 

(2)  N. 

(3)  S. 

(4)  S. 

(5)  S. 

(6)  N. 


15.50  chains ; 
25.40      " 


10.00 
19.70 
14.60 
21.00 


2. 


STA- 
TIONS. 

LINES. 

DlSTS. 

AZIMUTH  WITH 
LM. 

L 

LM 

22.45 

0° 

M 

MN 

1.30 

22° 

N 

NO 

66.30 

90° 

0 

OP 

23.85 

180° 

P 

PL 

67.10 

270° 

L 

LM 

360°  nr  0" 

DETERMINING   AREAS. 


193 


3. 

(1)  N.  llfE. 

(2)  S.     87°  W. 

(3)  S.  201°  W. 

(4)  S.     82°  E. 


13.19  chains; 

8.50      " 
11.75      " 
10.03      " 
AHS. 


acres. 


If  in  Article  76  we  substitute  respectively  tor  abscissa  and 
ordiiiate  of  a  corner  of  a  tract,  departure  and  latitude  of  the 
side  ending  at  said  corner,  the  rule  there  given  may  be  applied 
to  surveys  made  with  an  angular  instrument. 

To  illustrate,  take  the  example  given  on  page  191: 


CORRECTED 
LATITUDES. 

CORRECTED 
DEPARTURES. 

TOTAL 
LATITUDES. 

TOTAL 
DEPART- 
URES. 

DIFFER. 

BETWEEN 

ALTERNATE 
DEPARTS. 

DOUBLE 
AREAS. 

jr. 

S. 

E. 

W. 

1227 

468 

000 

9.06 

10.16 

-12.27 

4.68 

-14.84 

182.0868 

3.41 

.  . 

3.26 

-  3.21 

14.84 

-6.90 

22.1490 

3.34 

.  . 

3.35 

.20 

11.58 

6.61 

1.3220 

1.57 

.  . 

1.98 

3.54 

8.23 

5.33 

18.8682 

1.97 

6.25 

1.97 

6.25 

8.23 

16.2131 

2)240.6391 

10)  120  .32  sq.  ch. 

12.032  acres. 

In  this  case  the  axes  were  taken  through  the  most  westerly 
station,  thereby  making  the  total  departures  all  plus,  but  giving 
both  plus  and  minus  total  latitudes.  On  account  of  the  signs 
the  double  areas  are  all  plus.  The  axis  of  ordinates  passing 
through  the  most  westerly  station  makes  the  total  latitude  of 
that  station  zero,  and  consequently  there  is  one  less  multiplica- 
tion to  be  performed.  The  same  would  be  the  case  if  the  f 
axis  passed  through  the  most  easterly  corner. 


194 


PLANE   SURVEYING. 


Since  the  difference  of  the  alternate  total  departures  is  equal 
to  the  sum  of  the  adjacent  departures,  the  rule  just  given  may 
be  stated  as  follows  : 

Multiply  the  total  latitude  of  each  station  by  the  sum  of  the 
departures  of  the  adjacent  sides,  and  take  half  the  sum  of  these 
products  for  the  area. 

To  illustrate,  take  the  following  example  : 


S 

i 

J 

ii 

| 

BEARINGS. 

£ 

N. 

S. 

E. 

W. 

II 

I 

DOUBLF  AREAS. 

£ 

5 

3 

$« 

i 

N.  25°  E. 

433 

3PR 

183 

000 

2 

N.76'55'E. 

191 

43 

186 

393 

369 

145017 

3 

S.  63  41'  W. 

539 

535 

.  . 

62 

436 

124 

54064 

4 

S.  25°  W. 

40 

.  . 

36 

.  . 

17 

-  99 

-  79 

7821 

5 

N.  65°  W. 

320 

135 

290 

-135 

-307 

41445 

2)248347 

43560)  124173.5  sq.ft. 

2.852  acres. 

The'studeut  may  verify  the  preceding  example  by  this  method. 

2.  Given  the  bearings  and  distances  of  the  sides  of  a  field,  as 
follows,  to  find  the  area  by  each  of  the  two  preceding  methods. 
Ascertain,  also,  the  error  of  the  survey. 


(1)  N.    61°  W. 

(2)  N.  65. V°  E. 

(3)  S.     39°  E. 

(4)  S.       2°  W. 

(5)  S.     46°  W. 

(6)  N.    88°  W. 


9.38  chains; 
8.25  " 
6.51  " 
4.45  " 
5.00  " 
6.79  " 


8.   Given  the  boundaries  of  a  tract  of  land  with  the  corre- 
sponding weights,  as  follows,  to  determine  the  area  by  double 


DETERMINING   AREAS.  195 

meridian  distances,  using  the  weights  in  balancing  the  survey  as 
indicated  in  3°,  Article  208.  Determine,  also,  the  error  of  the 
survey. 

;i)   S.     79°  10'  W.,  dist.  27.00  chains,  weight,  1; 

"        3; 

"     H; 

»        2; 
2; 
"      2£; 

"      i; 
"      i. 

4.  The  distances  and  interior  angles  of  a  farm,  together  with 
the  bearing  of  one  line,  are  given  below.  The  angles  were 
measured  very  accurately.  It  is  required  to  calculate  the  area, 
by  either  of  the  preceding  methods,  balancing  the  survey  by 
(2°)  Second  Case,  Article  208.  Also  make  a  plot. 

Angle  L,       90° ;    side  LM,  28.00  chains. 
"      M,  148|°;     "    MN,25.2Q      " 


(1)  S. 

79°  10' 

W., 

dist.  27.00 

(2)  S. 

i° 

W., 

"     34.08 

(3)  N. 

89|° 

E., 

"     10.47 

(4)  N. 

1°55' 

E., 

"     15.30 

(5)  S. 

80|° 

E., 

"       7.15 

(6)  S. 

58£° 

E., 

"     11.50 

(7)  N. 

39° 

E., 

"       9.20 

(8)  N. 

16|° 

W., 

"     24.63 

ii 

N, 

81  1° 

;    " 

NO, 

11 

.70 

ii 

0, 

220° 

a. 

OP, 

12 

.IS 

tt 

P, 

90° 

a 

PQ, 

27 

.DO 

ii 

Q, 

90° 

a 

> 

QR, 

15 

.i<; 

« 

R, 

270° 

;    " 

RS, 

11 

.!)() 

ii 

S, 

90° 

>    " 

SL, 

21 

.(50 

Bearing 

of  LM,  N. 

10° 

E. 

5.  The  notes  of  a  survey  are  given  below ;  it  is  required  to 
determine  the  area  by  double  meridian  distances  after  correct- 
ing the  latitudes  and  departures  by  a  combination  of  2°  and  3°, 
Article  208.  See  also  note  in  same  article. 

The  interior  angles  were  observed. 

Angle  L,  91°  44';  side  LM,  17. 16  chains;  weight,  2. 

"      M,  168°  20';     "    MN,    9.48      "  "         1. 

"      N,  104°  49';     "    NO,     8.39      «  "       If 

'f       0,  179°  30';     "     OP,   15.28      "  "        2. 


196  PLANE   SURVEYING. 

Angle  P,  90°  19';  side  PQ,  16.05  chains;  weight,  2£. 

»      Q,  90°  05';     "     Q.R,  15.68      "              "           3. 

"      R,  283°  49';     "    RS,  11.40      "             "           1. 

"      S,  71°  24';     "    SL,  13.80      "             «           1. 

6.  Select  a  tract  of  land,  some  of  the  sides  being  much  more 
difficult  than  the  others  to  align  and  measure,  survey  it,  weight 
the  sides,  balance  the  latitudes  and  departures  according  to  the 
weights,  and  calculate  the  area. 

7.  Let  one  party  of  students  survey  a  tract  of  uneven  or  hilly 
land  of  considerable  magnitude,  by  means  of  transit  and  stadia 
and  rectangular  co-ordinates ;  another  party  at  the  same  time, 
or  the  same  party  subsequently,  survey  the  same  tract  in  the 
usual  way.     Compare  results. 

C.    WHEN  OFFSETS  ARE  TAKEN. 

235.  Let  the  annexed  figure  represent  the  case.  The  prop- 
erty lines  are  NO,  OP,  PL,  and  the  centre  of  the  creek  *  RS. 
Obtain  sufficient  data  to  compute  the  area  of  the  rectilinear 


— .0' 


*  When  a  non-navigable  stream  forms  a  boundary  of  a  tract  of  land, 
the  middle  of  it  is  considered  the  property  line,  unless  otherwise  specified. 
In  navigable  rivers  and  tidal  waters,  the  boundary  is  low-water  mark. 


DETERMINING   AREAS. 


197 


figure  LNOP,  and  take  offsets  from  the  line  LN  to  the  middle 
of  the  stream,  as  directed  in  Offsets  and  Tie-Lines,  Article  73  ; 
and  in  Traversing,  Article  164.  Calculate  the  area  of  LNOP 
by  one  of  the  preceding  methods ;  to  this  add  *  the  sum  of  the 
areas  of  the  trapezoids,  and  triangles  formed  by  the  offsets 
from  the  line  LN  to  the  middle  of  the  creek.  If  the  width  of 
the  stream  is  considerable,  and  especially  if  great  accuracy  is 
required,  the  surveyor  must  not  ignore  the  small  triangles  f 
formed  at  L  and  N.  O 

236.  To  Find  the  Area  of  a 
Pond  or  Small  .Lake,  traverse, 
or  take  the  bearings  of  the  sides 
LM,  MN,  NO,  etc.,  and  measure 
them  ;  also  take  offsets,  at  proper 
points,  from  these  lines  to  the 
edge  of  the  water. 

Calculate  the  area  included  be- 
tween the  right  lines,  and  sub- 
tract  therefrom  the  area  found  by 
the  offsets  ;  the  remainder  will  be 
the  area  required. 

EXERCISES. 

1.  Let  one  party  survey  a  field  with  compass  and  chain, 
taking  bearings  and  distances  of  all  the  sides  ;  another  party 
survey  the  same  field,  using  transit  and  chain,  and  observing 

*  If  the  base  line  LN  is  without  the  tract,  as  in  LNO'P1,  the  area 
included  between  the  middle  of  the  stream  and  LN  must  be  subtracted 
from  that  of  LNO'P'. 

t  Other  things  being  equal,  the  areas  of  these  small  triangles  depend 
upon  the  obliquity  of  PL  and  ON.  There  will  be  none  formed  when  PL 
and  NO  are  perpendicular  to  the  base  LN.  In  the  case  presented,  the 
area  of  the  triangle  at  L  is  to  be  added,  and  that  of  Nn'v  subtracted  from 
the  sum  of  the  areas  of  the  trapezoids,  to  obtain  the  correct  content 
between  LNund  the  middle  of  the  creek. 


198  PLANE   SURVEYING. 

the  interior  or  deflection  angles  ;  a  third  part}',  using  the  chain 
only.  Each  party  should  use  proof  lines,  make  record,  plot, 
and  calculate  the  area.  Compare  results. 

2.  With  a  transit,  survey  a  field,  a  part  of  which  is  bounded 
by  a  creek,  lake,  or  some  crooked  line  requiring  offsets  to  be 
taken  ;  make  a  plot,  and  compute  the  area. 

3.  Triangulate  a  portion  of  a  river  or  small  lake ;  make  a 
plot,  and  compute  the  area. 

4.  Make  the  necessary  measurements  to  write  a  description, 
to  make  a  plot,  and  to  compute  the  area  of  a  portion  of  a 
crooked  road. 

5.  Observe  all  the  bearings  and  measure  all  the  sides  of  a 
polygonal  field,  except  the  bearing  and  distance  of  one  side. 
Compute  the  area,  and  length  and   bearing  of  omitted  side. 
Subsequently  observe  the  bearing  and  distance,  and  note,  if  any 
disagreement,  how  much  the  area  is  affected  thereby. 


CHAPTER  III. 

DECLINATION  OF  THE  MAGNETIC  NEEDLE,  OE  VAKIATION 
OP  THE  COMPASS. 

237.  It  has  been   already  remarked  (Article  82)   that   the 
magnetic  and  geographic  meridian  do  not  in  general  coincide. 
The  angle  included  by  the  vertical  planes  containing  these  lines, 
or  the  angle  which  the  direction  of  the  needle  makes  with  the 
geographic  meridian,  is  the  declination  of  the  needle,  sometimes 
called  the  variation  of  the  compass.     It  is  different  at  different 
places,  and  is  a  variable  quantity  at  any  place. 

The  declination  is  termed  east  or  west,  according  as  the  north 
end  of  the  needle  points  to  the  east  or  west  of  the  geographic, 
or  true  meridian. 

The  magnetic  declinations  of  a  few  places  for  the  year  1885 
are  given  below : 

Eastport,  Me.,          19°  10' W.  Sitka,  Alaska,  28°50'E. 

Albany,  N.Y.,          10°11'W.  Milledgeville,  Ga.,       2°  32' E. 

Pittsburg,  Pa.,*          2°  52'  W.  New  Orleans,  La.,        6°  11'  E. 

Omaha,  Neb.,  10°  06' E.  City  of  Mexico,  Mex.,  7°  24' E. 

San  Francisco,  Cal.,  16°  34'  E. 

238.  Irregular  Changes.     The  magnetic  needle  is  subject  to 
disturbances  during  a  thunder  storm,  or  an  exhibition  of  aurora, 
solar  changes,  and  sometimes  it  is  considerably  agioated  with- 
out any  apparent  cause,  but  probably  on  account  of  magnetic 
or  electric  disturbances  more  or  less  remote. 

The  changes,  however,  which  especially  concern  the  surveyor, 
are  the  diurnal  and  secular. 

*  At  this  place,  September,  1887,  the  magnetic  declination  =  8°  01'  W. 


200  PLANE   SURVEYING. 

239.  The  Diurnal  Variation.  It  has  been  ascertained,  by 
repeated  observations  at  various  places,  that  the  magnetic 
needle  is  subject  to  daily  changes  ;  that  at  a  time  varying  from 
two  to  three  hours  after  sunrise  the  north  end  of  the  needle 
attains  its  maximum  deviation  to  the  east,  or,  as  it  is  called,  its 
eastern  elongation  ;  from  this  time  it  is  deflected  westward, 
attaining  its  western  elongation  between  1  and  2  o'clock  P.M., 
whence  it  retrogrades  towards  the  east.  There  is  sometimes 
an  interruption  of  the  motion  at  night,  but  generally  a  small 
reversed  movement  is  exhibited,  the  magnetic  meridian  being 
crossed  a  second  time  between  7  and  9  P.M.  The  times  at 
which  these  limits  are  reached  vary  with  the  seasons  :  during 
the  north  declination  of  the  sun  the  averages  for  eastern  and 
western  elongations,  respectively,  are  about  7.30  A.M.  and  1.15 
P.M.  ;  for  the  remainder  of  the  year,  about  8.45  A.M.  and  1.45  P.M. 

The  average  daily  direction  or  mean  magnetic  meridian  is 
reached  in  summer  about  10.15  A.M.,  and  in  winter  about 
10.45  A.M.,  at  Philadelphia,  and  generally  within  half  an  hour 
of  these  times  at  other  places. 

The  angular  range  between  these  limits  is  not  constant,  but, 
as  may  be  seen  by  the  table  subjoined,  it  is  considerably  greater 
in  summer  than  in  winter,  amounting  at  Philadelphia  to  10'  30" 
in  August,  and  only  6'  in  November,  or  a  yearly  average  of  8'. 
while  at  Key  West,  Florida,  the  average  for  the  year  is  about 
5'  30" ;  in  higher  magnetic  latitudes  the  average  being  more 
than  in  the  lower.  It  is  least  in  years  of  minimum  sun  spots 
(as  in  1878,  for  instance),  and  greatest  in  years  of  maximum 
sun  spots  (as  in  1870),  the  ratio  being  about  as  7  to  13  of  the 
average  amount  of  these  years  respectively.  The  daily  vari- 
ation is  at  times  interrupted,  at  others  enfeebled,  and  frequently 
in  the  winter  there  are  days  on  which  it  cannot  be  recognized. 
On  account  of  the  daily  movement  of  the  needle,  its  variable 
range  during  the  year,  and  disturbances  from  atmospheric  phe- 
nomena, it  is  well,  when  taking  the  bearing  of  any  important 
line,  to  record  the  date,  time  of  day,  and  condition  of  the 
atmosphere,  using  the  subjoined  table  as  far  as  practicable. 


VARIATION   OF   THE   COMPASS. 


201 


240.  For  reducing  the  direction  of  the  needle  observed  at 
other  hours  to  the  mean  magnetic  meridian,  the  following  table 
(taken  from  instructions  to  United  States  Deputy  Surveyors),  is 
furnished.  It  gives  to  the  nearest  minute  the  variations  of  the 
needle  from  its  average  position  during  the  day,  for  each  hour 
in  the  day,  for  the  four  seasons  of  the  year. 

TABLE  FOR  REDUCING  THE  OBSERVED  DECLINATION  TO  THE  MEAN 
DECLINATION  OF  THE  DAY. 


Hour 

6 

7 

8 

9 

10 

11 

12 

1 

2 

3 

4 

5 

6 

Spring 

3' 

4' 

fj 

3' 

1' 

V 

4' 

& 

5' 

4' 

3' 

2' 

1' 

Summer 

4' 

5' 

5' 

4' 

1' 

2' 

4' 

& 

5' 

4' 

y 

2' 

V 

Autumn 

2' 

3' 

3' 

2' 

0' 

2' 

3' 

4' 

3' 

2' 

i' 

I' 

0' 

\Vinter 

1' 

If 

2' 

2' 

1' 

0' 

2' 

3' 

3' 

2' 

i' 

1' 

0' 

241.  The  Secular  Variation.    Observations  extending  through 
many  years,  at  various  places,  indicate  a  continual  change  taking 
place  in  the  declination  of  the  needle ;  that  these  changes  are 
not  continuous  in  direction  nor  uniform  in  intensity  ;  that  in  this 
country  the  movement  which,  at  the  end  of  the  last  century,  was 
eastward  is  now  westward  at  all  places  east  of  the  Rocky  Moun- 
tains, and  that  a  period  of  250  or  300  years  may  elapse  before 
the  needle  will  again  resume  the  position  it  now  occupies.* 

242.  The  Line  of  no  Declination,t  or  Agonic  Line,  is  the  locus 
of  all  points  on  the  earth  where  the  direction  of  the  needle  is 

*  The  explanation  of  the  secular  change  must  ultimately  be  referred 
to  forces  of  a  periodic  character,  acting  for  centuries  with  great  regularity. 
So  far  no  approach  has  yet  been  made  towards  the  discovery  of  the  cause 
of  the  motion.  .  .  .  The  study  of  the  variation  of  the  declination  so.  far 
would  seem  to  indicate  a  secular  change  cycle  for  stations  in  the  United 
States,  extending  over,  or  varying  between,  the  limits  of  about  220  or  360 
years.  The  data,  however,  are  very  uncertain.  (U.  S.  C.  &  G.  S.,  1879.) 

t  Sometimes  called  the  Line  of  no  Variation. 


202  PLANE   SURVEYING. 

coincident  with  the  geographic  meridian.  At  all  places  on  the 
American  continent  situated  to  the  east  of  this  line  the  declina- 
tion is  west,  and  at  all  places  to  the  west  of  it,  the  declination 
is  east. 

The  line  of  no  declination  has  been  moving  westward  during 
the  present  century.  From  a  chart  published  by  Professor 
Loomis,  in  the  American  Journal  of  Science,  1840,  it  appears 
that  the  lines  of  equal  declination,  or  isogonic  lines,  crossed 
the  United  States  in  a  N.N.W.  direction  j  the  deflection  towards 
the  west  being  greatest  in  Maine.  The  line  of  no  declination 
at  that  time  entered  North  Carolina  about  midway  between 
Newbern  and  Wilmington,  passed  through  the  middle  of  Vir- 
ginia, and  into  Lake  Erie  at  a  point  nearly  equidistant  from 
Erie,  Pa.,  and  Cleveland,  Ohio. 

In  1885  the  Agonic  Line  entered  the  United  States  a  little  to 
the  east  of  Beach  Inlet,  S.C.,  thence  through  Greensboro,  N.C., 
Christiansburg,  Va.,  Point  Pleasant,  W.Va.,  St.  Clairsville, 
Ohio,  a  short  distance  west  of  Detroit,  and  a  few  miles  east  of 
Fort  Mackinac,  Mich. 

In  the  year  1700  the  declination  at  Philadelphia,  Pa.,  was  8|° 
west.  During  the  next  century  it  diminished,  reaching  a  mini- 
mum in  1800  of  IV  west,  since  which  time  it  has  been  increas- 
ing, and  is  now,  January,  1887,  at  the  Philadelphia  State  House, 
lat.  39°  56'  54",  long.  75°  09',  6°  50',  with  an  annual  increase 
of  5'. 

243.  Mr.  Charles  A.  Schott,  chief  of  the  computing  division 
of  the  U.  S.  C.  &  G.  S.,  tabulated  the  declinations  observed 
at  various  stations,  and  deduced  from  them  formulas  by 
which  the  magnetic  declination  at  various  places  may  be  com 
puted.* 

The  places  are  arranged  geographically  as  far  as  practicable, 
and  are  given  by  latitude  and  longitude  (west  of  Greenwich). 
The  epoch  to  which  the  formulas  refer  is  1850,  or  ra  =  t  —  1850. 

*  U.  S.  C.  &  G.  S.,  1882.     App.  12. 


VARIATION   OF   THE   COMPASS. 


203 


FORMULAS  EXPRESSING  THE  MAGNETIC  DECLINATION  AT  VARIOUS  PLACES 
IN  THE  UNITED  STATES,  AND  FOR  ANY  TIME  WITHIN  THE  LIMITS  OF 
OBSERVATION. 


NAME  OF  STATION  AND 
LOCATION. 

LATI- 

TUDK. 

LONGI- 
TUDE. 

EXPRESSION  FOR  MAGNETIC 
DECLINATION. 

Portland   Me        .... 

43°  38.8' 

70°  16.6' 

D—  +10?72  +  2?68  Bin(1.33m  +  24.1) 

Burlington,  Vt  

44°  28.2' 
43°  36.5' 

73°  12.3' 
72°  55.5' 

Z>=  +10.81  +  3.65  sin  (1.30  m-  20.5) 
+  0.18  sin  (7.0m  +  132) 
D—  +10  03  +  3  82  sin(l  am     243) 

Portsmouth,  N.H.    .    .     . 
Newburyport,  Mass.    .    . 

43°  04.8' 
42°  48.4' 
42°  31.  9' 

70°  43.0' 
70°  49.0' 
70°  52.5' 

D=  +10.63  +  3.17  sin(1.44m-4.7) 
D=  +10.07  +  3.10  sin(1.4m  +  1.9) 
D-+  9.80  +  3.61  sin  (1.50m  —  1.0) 

Boston,  Mass  '. 
Cambridge,  Mass.    .    .     . 

Nantucket,  Mass.     .     .    . 
Providence,  R.I  

Hartford,  Conn  
New  Haven,  Conn.  .    .    . 
\lbany   N  Y 

42°  21.5' 
42°  22.9' 

41°  17.0' 
41°  49.5' 

41°  45.9' 
41°  18.5' 
42°  39.2' 

71°  03.8' 
71°  07.7' 

70°  06.0' 
71°  24.1' 

72°  40.4' 
72°  55.7' 
73°  45  8' 

D=  +  9.52  +  2.93  sin(1.30  m  +  5.0) 
D=+  9.58  +  2.69  sin  (1.3m  +  7.0) 
+  0.18  ein  (3.2  m  +  44) 
D=+  9.29  +  2.78  sin(1.35m  +  5.5) 
D=+  9.10  +  2.99  sin  (1.45m  -3.4) 
+  0.19  sin  (7.2m  +  116) 
D=+  8.06  +  2.90  sin(1.25m  —  26.4) 
Z>=+  7.78  +  3.11  sin(1.40»i  —  22.1) 
D-+  8.17  +  3.02  sin  (1.  44  m  —  8.3) 

Oxford,  N.Y  
Buffalo,  N.Y  
Toronto,  Can  

Erie   Pa 

42°  26.5* 
42°  52.8' 
43°  39.4' 

42°  07.8' 

75°  40.5' 
78°  53.5' 
79°  23.4' 

80°  05.4' 

D  =  +  6.19  +  3.24sin(1.35m-18.9) 
D=+  3.66  +  3.47  sin(1.4m—  27.8) 
J)=+  3.  60  +  2.82  sin  (1.4m  -44.7) 
+  0.09  sin  (9.3m  +  136) 
+  0.08  sin(19m  +  247) 
D-+  2.26  +  2  71  sin  (1.55m  —  297) 

Marietta,  Ohio     .... 
Cleveland,  Ohio  .... 
Detroit,  Mich  
Sault  de  St.  Marie,  Mich. 
Cincinnati,  Ohio  .... 
St.  Louis,  Mo  
New  York,  N.Y.      .    .    . 

Hatborough,  Pa  
Philadelphia,  Pa.     .     .    . 
Harrisburg,  Pa  

39°  25.0' 
41°  30.3  ' 
42°20.0' 
46°  29.9' 
39°  08.6' 
38°  38.0' 
40°  42.7' 

40°  12.0' 
39°  56.9' 

40°  15.9' 
39°  17.8' 

81°  28.0' 
81°  42.0' 
83°  03.0' 
84°  20.1' 
84°  25.3' 
90°  12.2* 
74°  00.4' 

75°  07.0' 
75°  09.0' 

76°  52.9' 
76°  37.0' 

D=+  0.02  +  2.89  sin(1.4m  —  40.5) 
D=+  0.10  +  2.07  sin  (1.40m  —  6.2) 
Z>  =  —  0.97  +  2.21  sin  (1.50m  —  15.3) 
D=+  1.54  +  2.70  sin  (1.45m  -58.5) 
D  =  -  2.40  +  2.62  sin  (1.42  m  -  39.8) 
D  =  -  7.15  +  2.33  sin  (1.4m  -20.1) 
D=+  6.40  +  2.29  sin  (1.6m  -5.5) 
+  0.14  ain  (6.3m  +6.4) 
D=+  5.23  +  3.28  sin(1.54m-13.2) 
+  0.22  sin  (4.1m  +  157) 
Z>  =  +  5.38+  3.29  sin(1.55m-23.9) 
+  0.39  sin(4.0nt  +  161) 
D  =  +  2.93  +  2.98  sin  (1.50  m  +  0.2) 
D-+  3.20  +  2.57  sin(1.45m  —  21.2) 

Washington,  D.C.   .     .    . 
Cape  Henry,  Va.      .    .    . 
Charleston,  S.C  
Sa'-annah,  Ga  
Key  West,  Fla  

+  38°  53.3' 
+  36°  55.5' 
32°  46.6' 
32°  04.9' 
24°  33  5' 

+  77°  00.6' 
+  76°  00.5' 
79°  55.8' 
81°  05.5' 
81°  48.5' 

D=+  2.47  +  2.52  sin  (1.40m  -14.6) 
D=+  2.54  +  2.41  sin  (1.50m  -35.4) 
D=-  2.14  +  2.74  sin  (1.35m  -1.3) 
7>  =  —  2.54  +  2.32  sin  (1.5m  —  28.6) 
D  3.90  +  2.93  sin(1.4m  —  33.5) 

23°  09.3' 

82°  21  .5' 

D  4.52  +  2.00  sin(1.3m-  26.7) 

204  PLANE   SURVEYING. 

FORMULAS  EXPRESSING  THE  MAGNETIC  DECLINATION.  —  Continued: 


NAME  OF  STATION  AND 
LOCATION. 

LATI- 
TUDE. 

LONGI- 
TUDE. 

EXPRESSION  FOB  MAGNETIC 
DECLINATION. 

Kingston,  Jamaica  .     .     . 

17°  55.9' 

76°  50.6' 

D  =  —  4?64  +  2?04sin(1.2m  +  15°.9) 

Panama,  New  Granada    . 

8°  57.1' 

79°  32.2' 

D=-  6.80  +  1.82  sin(0.9  m  +  10.4) 

Florence,  Ala  

34°  47.2' 

87°  41.5' 

D  =  -  4.25  +  2.33  sin  (1.3  m  -  52.8) 

Mobile  Ala 

30°  41.4' 

88°  02.5' 

D=—  4.40  +  2.69  sin  (1.45  T/i  —  76.4) 

New  Orleans,  La.     .    .    . 

29°  57.  2' 

90°  03.9' 

D  =  —  5.61  +  2.57  sin(1.4»»  —  61.9) 

Vera  Cruz,  Mexico  .    .    . 

19°  11.9' 

96°  08.8' 

/>=-  4.38  +  5.04  sin  (1.10  TO  -65.0) 

Mexico,  Mexico   .... 

19°  25.9' 

99°  06.0' 

D=—  4.34  +  4.44  sin(1.0w  —  79.2) 

Acapulco,  Mexico    .     .    . 

16°  50.5' 

99°  52.3' 

D=-  4.13  +  4.82  sin(1.0»j  —  81.1) 

San  Bias,  Mexico     .    .    . 

21°  32.6' 

105°  15.7' 

D=-  6.51  +  2.  74  sin  (0.9  »«-  106.3) 

Magdalena  Bay,  L.  Cal.    . 

24°  38.4' 

112°  08.9' 

/)=  -  7.52  +  3.27  sin  (1.25  m  -140.6) 

San  Diego   Cal 

32°  42.1' 

117°  14.3' 

2>=  12.52  +  1.60  ein(l  2wi  179.8) 

Monterey,  Cal  

36°  36.1' 

121°  53.6' 

D  =  —12.90  +  3.28  sin(1.0  m  —  142.6) 

San  Francisco,  Cal.  .     .     . 

37°  47.  5' 

122°  27.2' 

D=  -13.34  +  3.23  sin(1.00  m  —  130.3) 

Cape  Disappointm't,  W.T. 

46°  16.  7' 

124°  02.0' 

D  =  —20.26  +  2.36  sin  (1.25  m  —  180.0) 

Sitka,  Alaska  

57°  02.9  ' 

135°  19.7' 

D  =  -26.77  +  2.33  sin  (1.4  m  -  111.6) 

Unalashka,  Alaska  .    .    . 

53°  52.6' 

166°  31.  5' 

D  =  -18.34  +  1.45  sin(1.4  m  -  67.8) 

Tyrone   Pa 

40°  40.0' 

78°  15.5' 

Z)=+  3.46  +  0.0550(£  —  1875.5) 

Pittsburg  Pa 

40°  27.6' 

80°  00.8' 

7)—  +  2.36  +  00566(£  1878.7) 

Chicago,  111  

41°  50.0' 

87°  36.7' 

Z>  =  -  6.03  +  0.0281  (t  —  1850) 

+  0.00082(<-1850)* 

Grand  Haven,  Mich.    .    . 

43°  05.2' 

86°  12.6' 

D  =  -  4.95  +  0.0380(<  —  1850) 

+  0.00120(<-  1850)2 

Madison,  Wis  

43°  04.6' 

89°  24.2' 

Z>=-  6.43  +  0.0655(^-1880.3) 

Duluth,  Minn.;  and  Supe- 

46° 45.5' 

92°  04.5' 

D  =  -10.17  +  0.0868(*  -  1875.8) 

rior  City,  Wis.      .    .    . 

Rio  Janeiro,  Brazil  . 

-22°  54.8' 

43°  09.5' 

Z>=+  0.282  +  0.1395(<  —  1850) 

+  0.00545  (<-  1850)  2 

San  Antonio,  Tex.  .    .    . 

+  29°  25.4' 

98°  29.3' 

D=  —10.14  +  0.0204(<  —  1850) 

+  .000024(<-  1850)2 

Omaha,  Neb.  ;  and  Council 

41°  15.7' 

95°  56.5' 

D=  -11.66  +  0.0439(*-1850) 

Bluffs,  Iowa  

Denver  Col 

39°  45.3' 

104°  59  5  ' 

2)  —  14.79  4-  0.0258(£  —  1872  9) 

Salt  Lake  City,  Utah.   .    . 

40°  46.1' 

111°  53.8' 

D  =  -15.51  -  0.0930(*  —  1850) 

+  0.00180(<-1850)2 

To  illustrate  the  use  of  the  table :  Suppose  it  is  desired  to 
ascertain  the  declination  of  the  needle  at  Harrisburg  for  the  last 
of  September,  1877,  or  t=  1877.75. 

Take  from  the  table  the  expression  for  the  declination  at 
Harrisburg  ;  that  is  : 

D=  +2.93 +  2.98  sin  (1.50m +  0.2). 


VARIATION    OF   THE   COMPASS.  205 

Find          m=  1877.75  -1850  =  27. 75; 

1.50ra  +  0.2  =  41.625  +  0.2  =  41.825, 
and        2.98  x  natural  sin  41.825  =  2.98  x  .66686  =  1.987. 

.-.  D  =  2.93 +  1.987  =  4.917  =  4°  55'  west  (the  result  being 
plus).  The  observed  declination  for  the  same  time  was  4°  53'  5". 
The  difference  between  the  computed  and  observed  declination 
is  seen  to  be  very  small. 

In  running  old  lines  it  may  be  necessary  to  determine  the 
declination  at  a  time  anterior  to  1850 ;  then  m  will  be  negative. 
Suppose  the  declination  at  Washington,  B.C.,  for  the  year  1841 
is  desired.  The  tabular  expression  is  : 

D  =  2.47  +  2.52  sin  (1.4m  -  14.6), 
m  —  1841  -1850  =  -9, 
(1. 4m  -14. 6)  =  -27.2, 
2.52  sin  (-27.2)  ==-1.15. 

.•.  D  =  2.47  — 1.15  =  1.32  west  (the  resulting  sign  beingpfots), 
which  agrees  practically  with  the  observed  declination. 

244.  The  following  table  is  taken  from  U.  S.  C.  &  G.  S.  Re- 
port, 1882,  App.  12,  Mr.  Schott's  paper  on  Secular  Variation. 
It  exhibits  the  computed  epoch  of  greatest  easterly  deflection 
reached  in  the  secular  motion  ;  i.e.,  the  date  when  last  reached, 
or  the  date  (in  parenthesis)  when  it  is  next  expected  to  be  in 
that  position  ;  the  amount  in  degrees  and  fractions,  and  direction 
(+  west,  —east)  at  this,  the  nearest  stationary  epoch  ;  and  the 
computed  annual  changes  in  tlie  declination  of  the  magnetic 
needle  for  the  years  1870,  1880,  and  1885,  a  plus  sign  indicating 
north  end  of  needle  moving  westward,  a  minus  sign  indicating 
north  end  of  needle  moving  eastward. 


206 


PLANE   SURVEYING. 


LOCATION. 

NEAREST 
STATIONARY  EPOCH 
OF  EASTERLY 
DIGRESSION. 

AMOUNT 
AT  EASTERLY 
DIGRESSION. 

ANNUAL  CHANGE. 

IN  1870. 

IN  1880. 

IN  1885. 

Paris  France 

1581 

—  10.6° 

—  7.0' 

—  6.1' 

—  9.5' 

Halifax,  Nova  Scotia    . 

1728 

+  12.4° 

+  1.8' 

+  1.0' 

+  0.5' 

Quebec,  Canada    .... 

1809 

+  12.1° 

+  4.2' 

+  1.6' 

+  0.5' 

Montreal,  Canada   .  .  . 

1816 

+    7.6° 

+  5.1' 

+  3.1' 

+  2.8' 

Eastport,  Me.  

1760 

+  12.5° 

+  3.3' 

+  2.7' 

+  2.3' 

Portland,  Me  

1764 

+   8.0° 

+  2.4' 

+  1.6' 

+  1.2' 

Burlington,  Vt  

1810 

+    7.2° 

+  5.0' 

+  6.0' 

+  5.8' 

Rutland,  Vt  

1806 

+   6.2° 

+  6.0' 

+  5.6' 

+  5.3' 

Portsmouth,  N.H.    .  .  . 

1791 

+    7.5° 

+  4.4' 

+  3.7' 

+  3.3' 

Newburyport,  Mass.  .  . 

1784 

+    7.0° 

+  3.9' 

+  3.3' 

+  2.9' 

Salem   Mass 

1791 

+   6.2° 

+  5.0' 

+  4.1' 

+  3.5' 

Boston,  Mass  

1777 

+   6.6° 

+  3.4' 

+  2.9' 

+  2.5' 

Cambridge,  Mass.   .  .  . 

1783 

+   6.9° 

+  2.9' 

+  2.1' 

+  1.8' 

Nantucket,  Mass.    .  .  . 

1779 

+   6.5° 

+  3.3' 

+  2.7' 

+  2.4' 

Providence,  R.I  

1780 

+   6.1° 

+  3.8' 

Hartford,  Conn  

1799 

+   5.2° 

+  3.8' 

+  3.7' 

+  3.6' 

New  Haven,  Conn.  .  .  . 

1802 

+   4.7° 

+  4.6' 

+  4.3' 

+  4.1' 

Albany,  N.Y  

1793 

+   5.2° 

+  4.3' 

+  3.7' 

+  3.4' 

Oxford,  N.Y  

1797 

+    3.0° 

+  4.5' 

+  4.3' 

+  4.0' 

Buffalo,  N.Y  

1806 

+   0.2° 

+  5.1' 

+  5.0' 

+  4.8' 

Toronto,  Canada  .... 

+  4.8' 

+  4.5' 

+  2.3' 

Erie,  Pa  

1811 

-  0.5° 

+  4.4' 

+  4.2' 

+  4.0' 

Marietta,  O  

1815 

-   2.9° 

+  4.2' 

+  4.2' 

+  4.2' 

Cleveland,  O  . 

1790 

—  2.0° 

+  2.8' 

+  2.5' 

+  2.2' 

Detroit,  Mich  

1800 

-   3.2° 

+  3.4' 

+  3.0' 

+  2.8> 

Sault  de  St.  Marie,  Mich. 

1828 

-   1.2° 

+  3.6' 

+  4.0' 

+  4.1' 

Cincinnati,  0            .  . 

1815 

—   5.0° 

+  3.8' 

+  3.9' 

+  3.8' 

St.  Louis,  Mo  

1800 

-   9.5° 

+  3.4' 

+  3.2' 

+  3.0' 

New  York,  N.Y  

1797 

+   4.0° 

+  2.4' 

+  2.5' 

+  2.6' 

Hatborough,  Pa  

1797 

+    1.8° 

+  4.6' 

+  4.5' 

Philadelphia,  Pa  

1800 

+    1.9° 

+  4.9' 

+  4.9' 

+  5.3' 

Baltimore,  Md  

1802 

+   0.6° 

+  3.9' 

+  3.6' 

+  3.2' 

Harrisburg,  Pa.     .... 

1700 

0.0° 

+  4.1' 

+  3.3' 

+  2.8' 

Washington,  DC.    .  .  .        1796 

0.0° 

+  3.5' 

+  3.2' 

+  3.0' 

VARIATION   OF   THE   COMPASS. 


207 


LOCATION. 

NEAREST 
STATIONARY  EPOCH 
OF  EASTERLY 
DIGKESSION. 

AMOUNT 
AT  EASTERLY 
DIGRESSION. 

ANNUAL  CHANGE. 

IN  1870. 

IN  1880. 

IN  1885. 

Cape  Henry,  Va  
Charleston,  S.C  

1814 
1784 
1809 
1810 
1801 
1762 
1739 
1821 
1841 
1830 
1827 
1839 
1841 
1868 
(1890) 
(1925) 
(1903) 
(1890) 
(1922) 
1865 
1834 

+  0.1° 
-  4.9° 
-  4.9° 
-  6.8° 
-  6.5° 
-  6.7° 
-  8.6° 
-  6.6° 
-  7.1° 
-  8.2° 
-  9.4° 
-  8.8° 
-  9.0° 
-  9.3° 
—  10.8° 
-14.1° 
-16.2° 
-  16.6° 
-22.6° 
-29.1° 
-19.8° 

+  3.8' 
+  3.5' 
+  3.6' 
+  4.3' 
+  2.7' 
+  2.0' 
+  1.5' 
+  2.8' 
+  2.8' 
+  3.1' 
+  4.2' 
+  2.4' 
+  2.4' 
+  0.1' 
-1.8' 
-1.8' 
-1.8' 
-1.0' 
-2.8' 
+  0.4' 
+  1.6' 

+  3.7' 
+  3.0' 
+  3.5' 
+  4.2' 
+  2.7' 
+  1.6' 
+  1.4' 
+  3.1' 
+  3.4' 
+  3.6' 
+  4.9' 
+  3.0' 
+  3.2' 
+  0.5' 
-1.0' 
-1.6' 
-1.3' 
-0.5' 
-2.5' 
+  1.2' 
+  1.9' 
+  3.3' 
+  3.4' 
+  4.6' 
+  6.6' 
+  3.9' 

+  5.2' 

+10.3' 
+  2.1' 

+  2.6' 

+  1.6' 
+  0.9' 

+  3.6' 
+  2.7' 
+  3.3' 
+  4.1' 
+  2.6' 
+  1.4' 
+  1.3' 
+  3.2' 
+  3.7' 
+  3.7' 
+  5.2' 
+  3.3' 
+  3.6' 
+  0.7' 
-0.5' 
-1.5' 
-1.0' 
-0.3' 
—  2.2' 
+  1.6' 
+  2.0' 

+  5.1' 
+  7.3' 

'  +10.7 
+  2.2' 

+  2.0' 

Key  West,  Fla  
Havana,  Cuba    
Kingston,  Jamaica  .  .  . 
Panama,  New  Granada 

Mobile  Ala              .  .  . 

New  Orleans,  La  
Vera  Cruz,  Mexico    .  . 
Mexico,  Mexico    .... 
Acapulco,  Mexico   .  .  . 
San  Bias,  Mexico    .  .  . 
Magdalena  Bay,  L.Cal. 
San  Diego,  Cal  

San  Francisco,  Cal.   .  . 
C.  Disappointm't,  W.T. 
Sitka  Alaska 

Unalashka,  Alaska  .  .  . 

Chicago    111 

1833 
1834 

-  6.3° 
-  5.3° 

.... 

Grand  Haven,  Mich.  .  . 
Madison   Wis     

Duluth,  Wis  ) 

Superior  City,  Wis.    .  > 
Rio  Janeiro,  Brazil    .  . 
San  Antonio,  Tex.  .  .  . 
Omaha,  Neb  ) 

Council  Bluffs,  la.  .  .  ) 
Denver   Col                .  . 

1876 

-16.7° 

.... 

Salt  Lake  City,  Utah    . 

208  PLANE   SURVEYING. 

The  variability  of  the  change  will  be  noticed.  For  example, 
take  New  York,  Philadelphia,  and  Harrisburg,  places  compara- 
tively near  together. 

At  New  York  the  change  in  1870  was  only  one-half  that  at 
Philadelphia ;  but,  both  increasing,  this  ratio  was  maintained 
throughout  the  15  years.  At  Harrisburg,  on  the  contrary,  the 
annual  change  in  1870  was  nearly  six-sevenths  that  at  Phila- 
delphia, but  the  change  constantly  increasing  at  the  latter  place 
while  diminishing  rapidly  at  the  former,  the  annual  variation  at 
Harrisburg  in  1885  was  only  a  little  more  than  one-half  that 
at  Philadelphia.* 

245.  Effects  of  the  Secular  Change.  It  is  evident  that  if  a 
surveyor  should  ignore  this  change,  in  attempting  to  establish 
the  corners  and  to  trace  the  boundary  lines  of  a  farm  from  their 
description  in  an  old  deed,  it  would  be  possible  for  him  to  return 
to  his  place  of  beginning,  but  probably  none  of  his  other  corners 
would  coincide  with  the  true  corners. 

A  line  in  the  vicinity  of  Philadelphia,  which  12  years  ago  had 
a  bearing  N.  19°  E.,  would  now  bear  N.  20°  E.,  and  in  the 
same  locality  a  bearing  which  at  that  time  was  recorded  N.  19° 
W.  would  now  be  N.  18°  W.  A  variation  which,  if  not  cor- 
rected, would  indicate  the  end  of  a  line  15  chains  long  over  26 
links  from  its  true  position. 

Take,  for  example,  the  notes  given  in  Article  208,  page  161, 
and  suppose  an  interval  has  elapsed  sufficient  to  make  the  vari- 
ation two  degrees.  The  accompanying  figure  shows  the  true 
lines  and  corners  ;  also  those  corresponding  to  a  survey  made 
without  taking  the  variation  into  account. 

The  bearings  and  distances  are  as  follows : 

(1)  S.   20°  53'    E.     13,11  chains; 

(2)  N.  48°  10'    E.     13.62      " 

(3)  N.  43°  40'  W.       4.73      " 

*  For  extended  investigations  on  magnetic  declination,  see  U.  S.  C.  & 
G.  S.  Reports,  1879,  1881,  and  1882. 


VARIATION   OF   THE  COMPASS. 

(4)  N.  45°  08'  W.  4.75  chains ; 

(5)  S.   51°  30'  W.  2.53      " 

(6)  S.   72°  30'  W.  6.56      " 


ss 


To  allow  for  a  variation  of  two  degrees,  we  should  have  the 
following  bearings : 

(1)  S.   18°  53'  E. ; 

(2)  N.  50°  10'  E. ; 

(3)  N.  41°  40'  W. ; 

(4)  N.  43°  08'  W. ; 

(5)  S.  53°  30'  W. ; 

(6)  S.   74°  30'  W. 

246.    To  deduce  a  general  rule  for  obtaining  the  magnetic 
bearings  of  old  lines  when  the  variation  is  known. 


210 


PLANE  SURVEYING. 


Let  JV$  represent  the  direction  of  the  magnetic  meridian  in 
the  vicinity  of  a  survey  made 
several  years  ago ;  N'S',  its 
direction  several  years  later,  at 
the  time  of  re-survey,  aud  that 
the  north  end  of  the  needle 
points  2°  farther  west.  It  is 
evident  that  at  the  time  of  the 
re-survey,  the  line  NS  will  bear 
N.  2°  E.,  and  OP,  which  accord- 
ing to  the  old  survey  bears  N. 
48°  E.,  will  have  its  bearing  in- 
creased 2°  or  N.  50°  E.  ;  but 
the  line  OM,  the  bearing  of  which  was  N.  42°  W.,  will  now  bear 
N.  40°  W.  A  line  recorded  as  east  will  be  traced  by  a  course 
S.  88°  E.,  and  soon. 

Hence  the  rule :  Increase  by  the  change  the  bearings  which 
are  northeasterly  or  southwesterly,  aud  diminish  by  the  same 
amount  the  bearings  which  are  northwesterly  or  southeasterly. 
The  foregoing  rule  is  directly  applicable  now  in  the  United 
States,  except  on  the  Pacific  coast,  because  the  variation  is  west. 
That  is,  the  north  end  of  the  needle  is  moving  west,  thereby 
increasing  the  readings  of  bearings  in  the  N.  E.  and  S.  W.  quar- 
ters, and  diminishing  the  readings  of  those  in  the  N.  W.  and 
S.  E.  quarters.  When  it  becomes  east,  the  words  "increase" 
and  "diminish"  should  be  interchanged  to  make  it  correct. 
If  a  vernier  compass  is  used,  the  variation  may  be  set  off  and 
the  lines  traced  by  the  old  bearings. 

247.  Change  Determined  by  Old  Lines.  If  the  bearing  and  date 
of  survey  of  a  line  are  known,  and  its  extremities  visible  from  each 
other,  setting  the  instrument  on  one  end  and  sighting  the  other 
will  give,  by  comparison  with  the  recorded  bearing,  the  variation. 

NOTE.  —  Care  must  be  taken  by  the  surveyor,  when  called  upon  to  run 
out  old  lines,  the  corners  not  being  definitely  marked,  that  the  time  of  the 
former  survey  be  known ;  the  date  of  the  deed  docs  not  indicate  that  of 
the  survey.  The  description  of  the  lines  may  have  been  copied,  as  they 
frequently  are,  from  an  older  deed. 


VARIATION   OF    THE   COMPASS.  211 

The  variation  to  be  applied  to  correct  magnetic  bearings  is 
frequently  determined  in  this  way. 

If  the  boundaries  of  a  tract  of  land  are  to  be  traced,  whether 
the  date  of  the  previous  survey  be  known  or  not,  the  surveyor 
seeks  to  find,  if  possible,  two  consecutive  marked  corners  ;  then, 
taking  the  bearing  of  these  and  comparing  with  the  record, 
he  obtains  the  change  sought. 

This  change,  properly  applied  to  each  side,  should  indicate 
its  direction. 

It  frequently,  and  in  large  tracts  generally,  happens  that 
though  the  corners  at  the  end  of  a  line  may  be  established,  they 
cannot  be  observed  from  each  other.  In  such  case  run  a  line 
as  nearly  as  possible  from  one  corner  towards  the  other  by  the 
bearing  given  in  the  deed,  or  make  first  an  allowance  which  may 
seem  proper  from  the  data  at  hand ;  measure  from  the  end  of  the 
line  thus  run  the  distance  to  the  true  corner,  and  by  the  57.3, 
rule,  Article  177  ;  or,  by  the  tangent  method,  same  article,  find 
the  angle  to  be  added  or  subtracted,  as  the  case  may  require,  to 
correct  the  bearing  with  which  to  run  the  line.  The  difference 
between  the  bearing  given  in  the  deed  and  the  corrected  bearing 
will  be  the  change  in  the  declination  since  the  survey  recorded 
in  the  deed. 

EXAMPLES. 

1.  A  line,  said  to  have  been  surveyed  in  1860,  recorded 
N.  18°  30'  E.,  24.40  chains,  was  run  in  1885  with  a  bearing 
N.  19°  45'  E., — the  variation  being  about  3'  to  the  west  per  year 
in  its  locality,  — and  the  corner  was  7  links  to  the  right  (farther 
easterly)  of  the  end  of  the  line  run.  The  corrected  magnetic 
bearing  and  variation  are  required. 

1°  15'  +  57;^  X  7  =  1°  15'  +  10'  =  1°  25'  =  variation. 

Adding  the  variation  to  the  bearing  of  the  line  run,  since  the 
true  corner  was  farther  to  the  east,  there  results  N.  19°  55'  E. 
as  the  corrected  magnetic  bearing  of  the  line. 


212 


PLANE   SURVEYING. 


2.  If  in  Example  1  the  corner  had  been  found  7  links  to  the 
left,  what  would  be  the  correct  bearing  of  the  line? 

3.  A  line  which  in  1862  ran  S.  34°  15' W.  18.56  chains,  in 
1886  bore  S.  35°  35'  W.      What  was  the  average  change  in 
the  declination  per  year? 

4.  Give  the  corrected  magnetic  bearing  for  1886  of  a  line  in 
the  same  locality  as  that  in  Example  3,  which  in  1868  ran  due 
east. 

5.  In  1876  a  line  had  a  bearing  S.  89°  45'  W.  16.80  chains  ; 
in  1886,  running  by  the  same  bearing,  the  true  corner  was  20 
links  to  the  north.     Give  the  average  annual  change,  and  cor- 
rect the  bearing. 

6.  If  a  line  60.00  chains  in  length  were  surveyed  in  the  early 
part  of  the  day,  where  the  needle  deviates  5  minutes  east  of 
the  mean  magnetic  meridian,  and  the  same  line  surveyed  soon 
after  mid-dav,  the  needle  then  pointing  5  minutes  west  of  the 
mean  magnetic  meridian,  how  far  apart  would  the  lines  be  at 
their  ends,  and  what  the  area  included  between  them? 

248.  To  Obtain  the  True  Bearing  of  a  Line,  that  is,  the 
bearing  with   respect  to  the  geographical  meridian,  when  the 


W 


declination  is  west.  Assume  NS  and  N'S'  (left-hand  figure)  to 
represent  respectively  the  true  and  magnetic  meridian.  Then  it 
is  evident  that  the  bearing  of  any  line  between  the  north  and 


VARIATION    OF   THE   COMPASS.  213 

east,  or  south  and  west,  as  OP  or  OP',  will  be  less  referred  to 
NS  than  when  referred  to  X'S'  by  the  amount  of  the  angle 
NON'  =  SOS'  =  the  declination. 

A  line  running  between  north  and  west,  as  OM,  or  south 
and  east,  as  OM',  will  evidently  have  its  bearing  increased  by 
the  amount  of  the  change. 

The  reverse  is  true  where  the  declination  is  east,  as  may  be 
perceived  by  reference  to  the  right-hand  figure. 

Hence,  to  get  the  true  bearing  from  the  magnetic  for  all 
places  east  of  the  line  of  no  declination,  i.e.  where  the  declina- 
tion is  tcest,  subtract  the  declination  from  a  bearing  which  is 
northeasterly  or  •  southwesterly,  and  add  the  declination  to  a 
bearing  which  is  northwesterly  or  southeasterly.  Where  the 
declination  is  east,  as  at  all  places  west  of  the  line  of  no 
declination,  add  the  declination  to  a  bearing  which  is  north- 
easterly or  southwesterly,  and  subtract  the  declination  from  a 
bearing  which  is  northwesterly  or  southeasterly.  Where  the 
declination  is  west,  a  bearing  that  reads  north,  when  reduced  to 
the  true  bearing,  will  evidently  be  west  of  north  the  amount 
of  the  declination  ;  if  the  declination  is  3°,  the  bearing  will  be 
N.  3°  W.,  and  supposing  the  same  declination,  a  line  running  due 
east  magnetically  will  be  truly  N.  87°  E. 

The  reverse  of  the  last  paragraph  is  true  where  the  declina- 
tion is  east. 

KKMARK.  If,  when  applying  the  rule,  a  negative  result  is  obtained,  oare 
must  be  exercised  in  the  interpretation  of  it.  For  example,  if  the  declina- 
tion is  3°  West,  and  the  needle  indicates  the  bearing  of  a  line  N.  1°  E., 
there  results,  by  the  rule,  —  2°.  This  shows  simply  that  the  true  bearing 
is  to  the  west  of  north,  or  N.  2°  W.  If  the  bearing  is  S.  89°  E.,  adding  the 
declination,  as  the  rule  requires,  gives  evidently  the  reading  N.  88°  E. 

Reduce  to  their  true  bearings  the  following,  the  declination 
being  2°  55'  W. : 

N.  2°  15'  E.,  East,  S.  45°  E.,  South ;  S.  87°  30'  W.,  N.  88° 
15'  W.,  North. 

Also  the  following,  the  declination  being   3°  40'  E.  : 

N.  88°  E.,  East,  S.  2°  E.,  South  ;  S.  88°  30'  W.,  N.  40°  W., 
North. 


214  PLANE   SURVEYING. 

249.  To  Ascertain  the  Declination.*    If  a  geographical  merid- 
ian were  traced  on  the  earth  convenient  to  the  operations  of 
the  surveyor,  he  would  have  the  means  always  at  hand  by  which 
to  determine  the  declination.     He  could  simply  set  up  his  in- 
strument at  a  point  on  the  meridian,  take  the  bearing  of  another 
point  in  it,  and  the  reading  would  be  the  declination.     So  the 
problem  resolves  itself  into  the  determination  of  a  geographic 
or  true  meridian. 

250.  By  Polaris.     If  there  was  a  celestial  object  precisely 
at  the  point  where  the  prolongation  of  the  earth's  axis  pierces 
the  celestial  sphere,  the  direction  of  the  meridian  could  be  ob- 
tained by  simply  sighting  to  the  object.     This,  however,  is  not 
the  case,  but  Polaris,  or  Alpha  Ursae  Minoris,  is  a  star  whose 
polar  distance  is,  January,  1887,  1°  17'  38", f  and  which  appar- 
ently revolves  about  the  north  pole  in  23  hours  56  minutes.     It 
therefore  culminates  twice  daily,  and  twice  it  attains  its  greatest 
distance  directly  east  and  west  of  the  pole,  called  respectively 
its  eastern  and  western  elongation.     If,  therefore,  the  Pole  Star 
could  be  observed  at  the  instant  of  its  culmination,  the  line  of 
sight  would  be  in  the  meridian  plane ;  but  since  in  general  the 
local  time  of  transit  is  not  precisely  known,  and  since  the  star 
is  then  moving  at  right  angles  to  the  plane  of    the  meridian 
respecting  which  its  motion  is  at  that  time  a  maximum,  and 
consequently  a  small  difference  in  time  would  introduce  a  con- 
siderable error  in  arc,  this  method  is  not  as  reliable  as  that 
by  means  of  Polaris  at  its  eastern  or  western  elongation,  as 
then  the  star  for  a  few  minutes  appears  to  move  in  the  direction 
of  the  vertical  wire,  or  compass-slit,  thus  affording  a  favorable 

*  For  other  methods,  see  Chapter  II.  Section  I,,  Solar  Attachment; 
and  Chapter  VI.,  Art.  Solar  Compass. 

t  Its  polar  distance  is  diminishing  at  the  rate  of  20"  (19.06")  per  year. 
This  diminution  will  continue  until  the  star  is  within  half  a  degree  of  the 
pole,  when  it  will  recede. 

In  1890  its  polar  distance  will  be  1°  16'  42". 

In  1900  its  polar  distance  will  be  1°  13'  33". 

In  1910  its  polar  distance  will  be  1°  10'  26". 


VARIATION   OF    THE   COMPASS.  215 

opportunity  for  observing  it,  and  the  precise  time  of  observation 
need  not  be  known. 

Conceive  a  spherical  triangle,  the  vertices  of  which  are,  Z, 
the  zenith  of  the  observer ;  P,  the  north  pole ;  and  S,  Polaris. 
This  triangle,  when  the  star  is  at  an 
elongation,  will  be  right-angled  at  the 
star.  In  this  right-angled  spherical 
triangle  are  known  the  co-latitude  of 
the  observer's  station,  and  the  co- 
declination  or  polar  distance  of  the 
star,  to  find  the  azimuth  *  and  hour 
angle. |  Using  natural  functions,  the 
formula  for  the  hour  angle  is  cos  P  =  tan  PS  cot  PZ,  and  for 
the  azimuth, 

5inZ=s[nPS  =  &[nP<S   + 
sinPZ      coslat   + 

It  may  be  well  to  remark,  though  it  has  only  a  theoretical 
significance,  that  these  formulas  are  not  applicable  to  all  north 
latitudes.  In  other  words,  there  will  be  no  hour  angle  shown 
by  the  first  formula,  nor  azimuthal  angle  by  the  second,  on  that 
parallel  of  latitude  which  agrees  in  arc  distance  with  Polaris 
from  the  equator,  and  for  any  point  between  that  parallel  and 
the  pole  the  formulas  fail. 

This  remark  is  in  general  applicable  to  any  circumpolar  star. 

QUERIES.  Is  Polaris  a  longer  time  passing  from  eastern  to 
western  elongation,  than  from  western  to  eastern,  to  an  ob- 
server whose  latitude  is  40°  ?  What  is  the  difference  in  time  to 
an  observer  whose  latitude  is  60°?  80°?  Where  would  this 
difference  be  a  minimum  ?  Where  a  maximum  ? 

*  The  azimuth  of  a  star  is  the  angle  between  the  meridian  plane  and 
the  vertical  plane  through  the  star. 

t  The  angle  SPZ  included  between  the  meridian  plane  PZ  and  the 
plane  PS  passing  through  the  star. 

}  The  azimuth  of  Polaris  at  elongation  varies  with  the  latitude  and 
with  the  year,  as  may  be  seen  by  the  table  on  page  217 


216 


PLANE   SURVEYING. 


251.  Table  of  mean  local  time  astronomical  (from  noon)  of 
the  elongations  and  culminations  of  Polaris  for  1885,  latitude 
40°,  and  longitude  6  hours  west  of  Greenwich. 


FIRST  DAY  OF 

E.  E. 

U.C. 

W.  E. 

L.  C. 

h.    m. 
0  353 

h.   m. 
6  29.9 

h.   m. 
12  24.6 

h.   m. 

18  28.0 

February  
March      

22  29.0 
20  38.5 

4  27.6 
2  37  1 

10  22.2 
8  31.8 

16  25.6 
14  351 

April  
May  

18  36.4 
16  38.6 

0  35.0 
22  33.3 

6  29.7 
4  31.8 

12  33.1 
10  35.2 

June    
July 

14  37.0 
12  39  5 

20  31.7 
18  342 

2  30.3 
0  32  8 

8  33.7 
6  362 

August  

10  38.1 
8  366 

16  32.8 
14  31  3 

22  27.5 
20  260 

4  34.8 
2  333 

October  

6  38.9 
4  37  0 

12  33.6 

10  31  7 

18  28.2 
16  264 

0  35.5 

22  29  7 

December   .  .  . 

2  38.9 

8  33.5 

14  282 

20  316 

To  correct  the  tabular  times  so  as  to  apply  to  any  year  subse- 
quent to  1885,  add  0.35  minutes  for  every  year.  For  any  year 
previous  to  that  date,  subtract  0.35  minutes  for  every  year. 

For  days  not  given  in  the  table,  interpolate,  or  allow  3.94 
minutes  for  each  day,  the  times  varying  by  this  amount. 

To  allow  for  difference  of  latitude  between  the  limits  of  30° 
and  50°,  add  0.14  for  every  degree  south  of  40° ;  subtract  0.18 
for  every  degree  north  of  40°. 

To  refer  the  tabular  times  to  any  year  in  a  quadriennium, 
observe  — 

For  the  first  year  after  a  leap  year  the  table  is  perfect ;  for 
the  second  year  after  a  leap  year  add  1  minute  ;  for  the  third 
year  after  a  leap  year,  add  2  minutes ;  for  a  leap  year,  and 
before  March  1,  add  3  minutes;  and  for  the  remainder  of  the 
year  subtract  1  minute. 


VARIATION   OF    THE   COMPASS. 


217 


It  will  be  noticed  that  there  occur  two  eastern  elongations 
on  Jan.  9,  and  two  western  elongations  on  July  9. 

AZIMUTH  (FROM  THE  NORTH)  OF  POLARIS,  WHEN  AT  ELONGATION,  BETWEEN 
THE  YEARS  1887-1895,  FOR  DIFFERENT  LATITUDES  BETWEEN  +26° 
AND  +50°. 


Lat. 

1887. 

1888. 

1889. 

1890. 

1891. 

1892. 

1893. 

1894. 

1895. 

+  25° 

1°25.7' 

1°25.3' 

1C25.0' 

1C24.6' 

1C24.3' 

1C23.9' 

1C23.6' 

1C23.2' 

1C22.9' 

26 

26.4 

26.0 

25.7 

25.3 

25.0 

24.6 

24.3 

23.9 

23.6 

27 

27.1 

26.8 

26.4 

26.0 

25.7 

25.4 

25.1 

24,7 

24.3 

28 

27.9 

27.6 

27.2 

26.8 

26.5 

26.2 

25.8 

25.4 

25.1 

29 

28.8 

28.4 

28.0 

27.6 

27.3 

27.0 

26.6 

26.3 

25.9 

30 

29.6 

29.3 

28.9 

28.5 

28.2 

27.8 

27.5 

27.1 

26.8 

31 

30.5 

30.2 

29.8 

29.4 

29.1 

28.8 

28.4 

28.0 

27.6 

32 

31.5 

31.2 

30.8 

30.4 

30.1 

29.7 

29.3 

29.0 

28.6 

33 

32.6 

32.2 

31.8 

31.4 

31.1 

30.7 

30.3 

30.0 

29.6 

34 

33.6 

33.3 

32.9 

32.5 

32.1 

31.8 

31.4 

31.0 

30.6 

35 

34.8 

34.4 

34.0 

33.6 

33.2 

32.9 

32.5 

32.1 

31.7 

36 

36.0 

35.6 

35.2 

34.9 

34.4 

34.0 

33.6 

33.2 

32.9 

37 

37.2 

36.8 

36.4 

36.0 

35.6 

35.2 

34.8 

34.5 

34.1 

38 

38.5 

38.1 

37.7 

37.3 

36.9 

36.5 

36.1 

35.7 

35.3 

39 

39.9 

39.5 

39.1 

38.7 

38.3 

37.9 

37.5 

37.1 

36.7 

40 

41.4 

41.0 

40.5 

40.1 

39.7 

39.3 

38.9 

38.5 

38.1 

41 

42.9 

42.5 

42.0 

41.6 

41.2 

40.8 

40.4 

40.0 

39.6 

42 

44.5 

44.1 

43.6 

43.2 

42.8 

42.4 

42.0 

41.5 

41.1 

43 

46.1 

45.7 

45.3 

44.9 

44.4 

44.0 

43.6 

43.2 

42.7 

44 

47.9 

47.5 

47.1 

46.6 

46.2 

45.8 

46.3 

44.9 

44.4 

45 

49.8 

49.4 

48.9 

48.5 

48.1 

47.6 

47.1 

46.7 

46.2 

46 

51.8 

51.3 

50.9 

50.4 

50.0 

49.5 

49.0 

48.6 

48.2 

47 

53.8 

53.4 

52.9 

52.5 

52.0 

51.5 

51.0 

50.6 

60.2 

48 

56.0 

55.6 

55.1 

54.6 

54.2 

53.7 

53.5 

52.8 

52.3 

49 

58.3 

57.9 

57.4 

56.9 

56.5 

56.0 

55.5 

55.0 

64.5 

+  50° 

2°00.8' 

2°00.3' 

1°59.8' 

T59.3' 

1°58.8' 

F68.4' 

F67.9' 

1°67.4' 

1°66.9' 

218  PLANE   SURVEYING. 

252.  To  Establish  a  True  Meridian  with  a  Transit.*  See 
that  the  instrument  is  in  good  adjustment.  Allow  sufficient 
time  before  an  elongation  of  the  star  to  "set  up"  the  transit 
in  a  desirable  position,  f  See  that  it  is  planted  firmly,  levelled 
carefully,  and  that  the  cross-wires  are  illuminated  J  and  prop- 
erly focused.  For  convenience,  set  the  vernier  at  zero,  and 
unclamp  the  lower  plate. 

Observe  the  star  a  few  minutes  before  its  elongation,  and 
keep  the  vertical  wire  on  it  by  clamping  the  lower  plate  and 
using  the  slow-motion  screws  attached  to  it.  When  it  has 
attained  its  greatest  elongation,  it  will  appear  for  a  few  moments 
to  coincide  with  the  vertical  wire,  and  then  retrograde.  Un- 
clamp the  vernier  plate,  and  turn  off  with  it  the  amount  of  the 
azimuth  §  corresponding  to  the  time  and  place  as  given  in  the 
table  of  the  preceding  article.  The  telescope  will  then  point 
in  the  direction  of  the  true  meridian,  and  a  mark  should  be  set 
at  as  long  range  as  practicable.  If  preferred,  a  stake  may  be 
set  in  line  of  sight  at  elongation,  leaving  the  turning  off  of 
azimuth,  and  setting  mark  in  meridian  until  the  next  day.  It 
would  be  a  little  more  accurate  to  take  the  mean  of  several 
observations  —  direct  and  reverse  —  at  eastern  and  western 
elongations. 

*  See  Solar  Attachment,  Chapter  II.  Section  I. ;  also  Solar  Compass, 
Chapter  VI. 

t  Twenty  to  thirty  minutes  usually,  depending  upon  the  observer. 

t  Perforated  silvered  reflectors,  for  this  purpose,  can  be  obtained  of 
instrument  makers.  Or,  cover  with  white  paper  a  board  12  or  15  inches 
square,  make  a  perforation  through  it  of  2  or  3  inches'  diameter,  and  nail 
on  a  piece  of  board  to  hold  a  candle.  This  reflector  may  be  attached  to 
a  staff,  that  it  can  slide  up  and  down,  and  adjusted  to  the  height  of  the 
telescope.  It  should  be  placed  about  a  foot  from  the  object-glass,  so  that 
the  reflection  from  the  paper  will  render  the  cross-wires  visible,  and  at 
such  a  height  that  the  star  can  be  observed  through  the  opening. 

§  The  meridian  will  lie  to  the  west  or  east  of  the  direction  of  the  tele- 
scope when  elongation  was  observed,  according  as  the  elongation  was  east 
or  west.  The  azimuth  must  be  turned  off  accordingly.  Since  the  direc- 
tion of  the  line  from  the  observer's  station  to  the  star  at  elongation  is 
known,  the  declination  may  be  ascertained  even  before  the  meridian  is 
established. 


VARIATION    OF   THE   COMPASS.  219 

253.  The  Direction  of  the  Meridian  may  be  found,  though 
less  accurately,  bv  means  of  a  compass-sight  and  plumb-line. 

Take  a  smooth  plank  about  3  feet  in  length,  and  fix  it 
firmly  level,  and  nearly  east  and  west,  on  supports  about  2  feet 
high.  Attach  a  compass-sight  to  a  board  6  or  8  inches  square. 
At  15  or  20  feet  n9rth  of  the  plank  suspend  a  plumb-line  by 
artificial  supports,  from  some  projecting  point  on  a  building  or 
at  the  end  of  a  staff  projecting  from  a  high  window. 

At  fifteen  or  twenty  minutes  before  the  time  of  elongation  of 
the  star  let  an  assistant  hold  a  light  in  such  position  that  the 
plumb-line  may  be  distinctly  seen  through  the  compass-sight 
when  placed  on  the  plank.  Move  the  sight  until  the  plumb- 
line  covers  the  star.  Continue  to  keep  the  star  and  line  in  that 
relative  position  until  the  star  begins  to  retrograde.  The  direc- 
tion of  the  line  of  sight  then  corresponds  to  that  observed  by 
the  transit  as  indicated  in  the  preceding  article ;  and  applying 
the  azimuth  therein  directed,  the  meridian  may  be  set  out.* 

254.  To  Obtain  approximately  the  Meridian.     In  old  works 
on  surveying  it  is  stated  that  the  north  star  (Polaris)  is  very 
nearly  the  meridian  when  it  and  Alioth  f  are  in  the  same  vertical 
plane  or  line.     Others  add  the  time  that  must  elapse  after  one 
is  vertically  above  the  other  before  the  north  star  makes  its 
transit,  and  then  by  sighting  the  north  star  at  that  instant  the 
meridian  may  be  found. 

This  interval  is,  January,  1887,  nearly  half  an  hour.  Other 
stars  are  now  used,  being  more  suitable.  Zeta,  or  Mizar,  the 
star  next  to  Alioth  in  the  tail  of  the  Great  Bear,  comes  to  the 
meridian  now  almost  simultaneously  with  Polaris  and  at  a  con- 
venient time  in  the  autumn  and  early  winter  to  make  the  obser- 

*  If  possible,  a  night  should  be  chosen  when  there  is  no  wind.  The 
slightest  disturbance  in  the  air  causes  considerable  vibration  of  the  plumb- 
line.  Using  a  heavy  "  bob,"  and  allowing  it  to  vibrate  in  a  vessel  of  water, 
will  tend  to  the  accuracy  of  the  result. 

t  Alioth,  or  Epsilon:  the  star  in  the  tail  of  the  Great  Bear  nearest  the 
quadrilateral. 


220  PLANE   SUKVEYLNG. 

vation.  Delta  Cassiopeiae,  which  is  on  the  same  side  of  the 
pole  as  Polaris,  makes  its  transit  also  about  the  same  time  with 
it,  and  may  be  used  in  the  spring  and  early  summer  when  it  is 
not  practicable  to  make  use  of  Zeta.  To  make  either  of  these 
observations,  use  a  transit,  or  a  plumb-line  and  compass-sight, 
as  explained  in  the  preceding  articles ;  watch  the  movements 
of  the  stars  until  they  coincide  with  the  plumb-line.  The 
direction  of  the  line  of  sight  then  will  indicate  quite  closely  the 
meridian.* 

*  The  vertical  plane  including  Zeta  and  Polaris  is  slowly  moving  east- 
ward at  about  the  rate  of  two  minutes  in  six  years.  At  the  present  time 
(1887)  Polaris  is  on  the  meridian  about  two  minutes  before  Zeta  of  the 
Great  Bear,  but  in  six  years  their  respective  upper  and  lower  transits  will 
coincide.  The  vertical  plane,  including  Delta  Cassiopeia?  and  Polaris,  is 
moving  westward  at  about  the  same  rate.  Polaris  now  comes  to  the 
meridian  about  one  minute  before  this  star. 


CHAPTER  IV. 


LAYING  OUT  AND   DIVIDING  LAND. 

SECTION   I. 
LAYING  OUT  LAND. 

A.    TRIANGLES. 

255.  To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
triangle  when  the  length  of  the  base  is  given. 

Denote  the  given  area  in  square  chains  or  square  rods  by 
A,*  the  length  of  the  base  (referred  to  the  same  unit)  by  6, 
and  the  unknown  altitude  by  x.  Then 

AT  9  A  M  P'  P          K 

-  =  ^L,  oro?  =  —   Measure  the  base,    ' 

and  at  any  point  in  it  erect  a  perpen- 

2  A 

dicular   equal   to  — •      Join   the   ex-    0 
b 

tremity  of  the  perpendicular  with  the  extremities  of  the  base, 
and  a  triangle  fulfilling  the  conditions  of  the  question  will  be 
exhibited. 

256.  Wlien  the  area  is  given  and  the  base  and  altitude  in  a 
given  ratio. 

*  Why  not  let  A  denote  the  number  of  acres  ? 

NOTE.   The  locus  of  the  vertices  of  the  triangles  answering  the  conditions 

O    A 

is  a  line  parallel  to  the  given  base  and  at  a  distance  therefrom  = • 

o 


222  PLANE   SURVEYING. 

Designate,  as  before,  the  ai'ea  by  A,   the  base  and  altitude 
respectively  by  x  and  y,  and  -  =  —  the  ratio ;  then 


Or,  let  mx  =  base  and  nx  =  altitude  ;  then 


whence 


and  —        l'2Am 


•2  An 


257.    Given  area,  6ase,  and  one  side,  to  make  a  given  angle 
with  the  base. 

Denote  the  base  and  area  as  above ; 
then,  since 

PN=  OP  sin  0, 

6  X  OP  sin  0 


and 


0  N  R  QP  = 


2A 
b  sin  0 


EXAMPLES. 

1.  Lay  out  an  isosceles  triangle  to  contain  6  acres,  making 
the  base  |  the  altitude.     Locate  the  altitude  and  find  its  length. 

2.  Lay  out  a  right  triangle  containing  4  acres,  having  the 
base  |  the  altitude. 

3.  It  is  required  to  lay  out  2  acres  in  the  form  of  a  triangle, 
the  base  to  be  7.50  chains.     Find  the  length  of  a  side  of  this 
triangle  which  shall  make  an  angle  of  40°  with  the  base. 


LAYING    OUT   AND   DIVIDING    LAND.  223 

258.  To  lay  out  an  equilateral  triangle  to  contain  a  given  area. 
Let  x  =  the  side,  and  A  =  the  area ;  then,  since 


.433 

259.  Given  the  area  and  the  two  sides,  to  lay  out  the  triangle. 
Denote  the  given  sides  by  b  and  c,  the  area  by  A,  and  the 

nknown  angle  by  a;  then,  since 

'I  sin  a  =  A, 

=  ~bc" 

EXAMPLES. 

1.  Find  the  side  of  an  equilateral  triangle  containing  one 
acre. 

2.  What  is  the  altitude  of  the  triangle  in  Example  1  ?    How 
far  is  it  trom  the  foot  of  the  perpendicular  to  the  centre  of  the 
figure?     How  far  from  either  angle  to  the  centre? 

3.  Lay  out   a  triangle  containing  2  acres,  two  sides  to  be 
8  chains  and  6  chains.     What  must  be  the  included  angle? 

B.     QUADRILATERALS. 

SQUARES. 

260.  To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
square. 

Denote  the  required  area  in  square  chains  or  square  rods 
by  -A,  and  one  of  the  sides  by  x ;  then  x  =  -\J  A. 

Measure  a  distance  equal  to  the  V2. ;  at  each  extremity  of 
this  line  erect  a  perpendicular  of  the  same  length ;  connect  the 
extremities  of  the  perpendiculars  ;  the  figure  will  be  a  square. 


224  PLANE    SURVEYING. 

RECTANGLES. 

261.  To  lay  out  a  given  quantity  of  land  in  the  form  of  a  rec- 
tangle, one  side  being  given. 

Denote,  as  before,  the  area  by  A,  the  given  side  by  6,  and 
by  x  the  unknown  side  ;  then 

.=4 

b 

262.  Given  the  area,  and  the  length  to  the  breadth  in  a  given 
ratio. 

Denote  the  area  as  above  ;    the  length  and  breadth  respec- 
tively by  x  and  y  ;  m  and  n  their  ratio,  so  that 

x  _  m 

y~  n 
Then,  since  xy  =  A,  there  results,  by  substitution, 

-V- 


Or,  let  mx  =  the  length,  and  nx  —  the  breadth  ;  then 
mnx2  =  A  ; 
Am 


whence 

n 


and  nx  =\|— . 

*   tn. 


An 
m 

263.    Given  the  area  and  the  sum  of  the  length  and  breadth. 
Denote  the  sum  of  the  sides  by  S ;   the   other    notation  as 
above ;  then 

and  x  +  y  =  S ; 

whence 

and  y  = 


LAYING   OUT    AND   DIVIDING   LAND.  225 

264.  Given  the  area  and  the  difference  of  the  length  and 
breadth. 

Denote  the  difference  of  the  sides  by  d ;  the  other  notation  as 
before  ;  then 


whence 


and 


EXAMPLES. 


1.  How  many  rods  in  each  side  of  a  square  lot  which  con- 
tains 1  acre  ?     How  many  chains  ?     How  many  yards  ? 

2.  Lay  out  6  acres  in  the  form  of  a  rectangle,  the  length  of 
one  side  to  be  10  chains.     Find  the  adjacent  side. 

3.  Find  the  sides  of  a  rectangle  which  shall  contain  15  acres, 
and  the  length  ^  the  breadth. 

4.  It  is  required  to  lay  out  a  rectangle  containing  12  acres, 
so  that  the  sum  of  two  adjacent  sides  shall  equal  26  chains. 
What  must  be  the  length  and  breadth  ? 

5.  Find  the  sides  of   a  rectangle  which  shall  contain  640 
square  rods,  and  the  difference  of  whose  sides  is  10  rods. 

PARALLELOGRAMS. 

265.  To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
parallelogram,  the  base  being  given. 

Denote  the  area  and  base,  as  above,  and  the  altitude  by  x  ;  then 
A 

—T 

From  any  point  in  the  base  erect  a  perpendicular  equal  to 
A  -j-  6,  and  through  the  extremity  of  the  perpendicular  run  a  line 
parallel  and  equal  to  the  base:  a  parallelogram  will  thus  be 
formed,  fulfilling  the  conditions  of  the  question. 


226  PLANE    SURVEYING. 

266.    Given  the  area,  one  side,  and  adjacent  angle. 
Denote  the  area  by  A,  the  base  by  6,  the  given  angle  by  a, 
p  M   and  by  x  the  side  adjacent ;  then 

bx  sin  a  =  A. ; 
A 


whence 


b  sin  i 


L  N 

Turn  off  at  L  and  N,  the  given  angle,  measure  the  dis- 
tances LP  and  NM,  equal  x,  and  connect  M  and  P  for  the  de- 
sired figure. 

267'.  Given  the  area  and  two  adjacent  sides,  to  find  the  included 
angle. 

Denote  the  sides  by  b  and  c,  their  included  angle  by  a,  and 
the  area  as  above  ;  then 

be  sin  a  =  A ; 

* 

whence  sin  a  =  — 

be 

QUERIES.  What  will  the  figure  become  when  be  =  A? 
When  b  =  c?  May  the  product  of  be  be  less  than  A?  Can  an 
expression  for  the  sine  be  obtained  for  each  case  ? 

EXAMPLES. 

1.  It  is  required  to  lay  out  a  parallelogram  to  contain  200 
square  rods,  having  a  base  of    20  rods.     What  must  be  the 
altitude  ? 

2.  If  in  Example  1  it  is  required  that  the  perpendicular  shall 

M  be  erected  at  the  middle  of  the  base, 
and  terminate  at  the  angle  P,  as 
per  figure,  what  length  must  be 
given  LP,  and  what  the  magnitude 


£ -J  of  the  angle  L? 

3.  It  is  required  to  lay  out  a  parallelogram  to  contain  48 
square  chains,  one  side  to  be  8  chains,  and  the  adjacent  angle 
70°.  What  must  be  the  length  of  the  adjacent  side? 


LAYING    OUT   AND   DIVIDING   LAND.  227 

4.  It  is  required  to  lay  out  a  parallelogram  to  contain  2.4 
acres,  the  base  and  adjacent  side  to  be  respectively  6  and  5 
chains.    Determine  the  altitude  and  tell  how  to  lay  out  the  land. 

5.  It  is  required  to  lay  out  a  rhombus  to  contain  32  square 
chains,  each  side  to  be  6  chains.     Compute  the  altitude,  and 
state  how  to  set  out  the  tract  ;  that  is,  to  establish  every  corner. 

C.   POLYGONS. 

268.    To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
regular  polygon  of  any  number  of  sides. 

Denote  the  area  by  A,  the  number  of  the  sides  by  n, 
and  the  length  of  one  of  the  sides,  as  PN  in  the  p  L  ff 
figure,  by  #,  and  ON,  the  radius  of  the  circum- 
scribed circle,  by  y  ;  then 

"n  X  OL  X  LN=  A. 


and 


But  the  angle  LON=,  OL=     cot 


Whence          x  = 

and  y  = ^ 

n 

To  lay  out  the  tract,  find  by  the  above  formula  the  length  of  one 
side,  as  LN,  and  stake  it  out.  Then  with  an  instrument  for  meas- 
uring angles  (transit)  set  up  at  one  end,  as  N,  sight  L,  plunge 

the  telescope,  deflect  — - —  to  M.     Measure  NM=  NL.     Remove 

n 

the  instrument  to  M,  deflect  from  the  prolongation  of  MN,  as 
before, ,  measure  MP,  and  so  continue  around,  locating  PQ, 


228  PLANE   SURVEYING. 

and  finally  returning  to  L.     The  figure  will  be  the  polygon 
required. 

In  a  small  polygon,  if  the  centre 
is  fixed,  it  will  be  better  to  set  up 
on  it  and  measure  therefrom  a  dis- 
tance y  to  N,  turn  off  an  angle  (the 

instrument  still  at  the  centre)  = , 

and  measure  the  same  distance  to  M, 
again  turning  off  an  angle  equal  to 
N  L  the  last,  measure  the  same  distance 

to  P,  and  so  on.     A  stake  planted  at  each  extremity  of  the 
radial  lines  will  indicate  the  angular  points  of  the  tract.* 

EXAMPLES. 

1.  Show  how  to  laj*  out  1210  square  yards  in  the  form  of  an 
octagon.     The  same  for  a  pentagon  ;  decagon. 

2.  Show  how  by  Article  57  the  length  of  a  side  of  a  polygon 
of  a  give-n  area  and  any  number  of  sides,  within  the  limits  of 
the  table,  may  be  found. 


D.   CIRCLES  AND   ELLIPSES. 
CIRCLES. 

269.   To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
circle. 

Denote   the  area  by  A,  and  the  radius  by  x  ;   then,  since 

rj 


*  A  small  lot,  when  great  accuracy  is  not  required,  may  be  laid  out 
by  fastening  one  end  of  a  tape  at  0,  and  with  a  length  ON  mark  out  a 
circumference  by  means  of  a  pin.  Then,  beginning  at  any  point  in  the 
circumference,  measure  off  the  distance  x,  and  continue  round  the  curve. 
driving  a  stake  at  the  extremity  of  each  side. 


LAYING   OUT   AND   DIVIDING  LAND.  229 

When  great  accuracy  is  not  required,  and  small  circles  gener- 
ally may  be  laid  out  by  fastening  one  end  of  a  tape  at  the 
centre,  and  with  a  common  marking-pin  held  firmly  and  perpen- 
dicularly along  it  at  x  distance,  describe  and  mark  out  the 
circumference. 

270.  Or,  fix  the  extremities  of  two  diameters  run  out  per- 
pendicular to  each  other,  connect  these  with  chords,  and  the 
versed  sine  of  45°  to  the  known  radius  will  give  at  once  the 
perpendicular  distance  from  the  centre  of  each  chord  to  the  cir- 
cumference.     If  necessary,  the  points  thus   located   may  be 
connected  and  others  found  in  a  similar  manner.     Or  the  per- 
pendicular distance  from  any  given  point  in  a  chord,  of  known 
length,  to  the  circumference  may  be  found  by  simple  geomet- 
rical truths  deduced  from  the  right  triangle. 

271.  If  the  circle  is  too  large  to  be  laid  out  as  above,  it  may 
be  accomplished  by  means  of  deflec-      L 

tion  angles  as  follows :  With  the 
known  radius  find  the  angle  at  the 
centre  (7,  which  is  subtended  by  a 
chord  OM  of  any  length,  say  100 
feet ;  then  with  the  instrument  at  M, 
deflect  from  the  tangent  ML  to,  0  an 
angle  LMO  =  one-half  the  central 
angle  OCM,  and  measure  the  distance 
3/0=  100  feet.  0  is  a  point  in  the 
curve.*  Again  deflect  an  angle  OMP  =  one-half  the  central 
angle,  and  measure  OP  =  100  feet  to  locate  /*,  another  point  in 
curve,*  and  so  on  to  locate  the  others.  If  there  is  a  fractional 
part  of  the  deflection  angle  at  the  closing  point,  the  correspond- 
ing fractional  part  of  100  feet  may  be  used. 


*  The  angle  formed  by  the  tangent  and  chord  drawn  to  the  point  of 
contact  is  measured  by  one-half  the  intercepted  arc.  An  inscribed  angle 
has  the  same  measure. 


230 


PLANE   SURVEYING. 


ELLIPSES. 

272.  To  lay  out  a  given  quantity  of  land  in  the  form  of  an 
ellipse,  the  greater  and  lesser  diameters  to  bs  in  a  given  ratio. 

Denote  the  area  by  A,  the  greater  and  less  diameter  (axes) 
respectively  by  mx  and  nx,  in  which  m  and  n  express  the  given 
ratio;  then,. 


'->££ 


and 


273.    Given  the  area  and  one  of  the  diameters,  to  find  the 
other  diameter. 

Denote  the  given  diameter  by  (Z,  the  unknown  by  x,  and  the 
area  as  before  ;  then,  since 


we  have 


An  ellipse  of  small  size  may  be  laid 
out  as  follows  : 

Measure  AB  equal  to  the  greater  diameter  (transverse  axis)_, 
and  from  the  centre  0  lay  off  OF=  OF',  each  equal  to  the 
square  root  of  the  difference  of  the  squares  of  the  semi- 
diameters  OA,  OC.  Fix  the  ends  of  a  steel  wire  or  ribbon  of 
the  length  AB  at  F  and  F1  ',  and  with  a  continuous  motion  of  a 
marking-pin  P,  held  perpendicularly,  keeping  the  wire  taut,  the 
required  curve  will  be  traced. 

*  See  any  work  on  General  Geometry  or  Conic  Sections  for  the  area  of 
an  ellipse. 


LAYING    OUT    AND   DIVIDING    LAND.  231 

Or,  having  found  the  axis  as  above,  P  being  any  point  in 
the  curve,  and  PR  perpendicular  to  AE  at  R,  by  setting  off 
any  number  of  points  on  AB,  we  may  find  from  the  proportion 

TR2 :  RB  x  AU  =  OC* :  Od2, 
the  corresponding  values  of  PR. 

EXAMPLES. 

1.  Find  the  radius  of  a  circle  containing  1  acre. 

2.  Find  the  radius  of  a  sector  containing  20  square  rods, 
the  angle  at  the  centre  being  72°. 

3.  The   area  of  an   ellipse  is   1  acre,  its  diameters  in  the 
ratio  of  3  :  2  ;  find  their  length. 

4.  An  ellipse  contains   80   square   rods,  its   greater   diam- 
eter 12  rods  ;  find  the  lesser  diameter. 

5.  The  greater  diameter  of  an  elliptical  plot  of  ground  en- 
closed by  a  wall  1  foot  thick  is  240  links,  and  the  lesser  160 
links,  inside  measurements.     What  is  the  area  of  the  plot,  and 
how  much  land  is  occupied  by  the  wall  ? 

274.  Let  it  be  required  to  lay  out  a  circle  circumscribing  a 
triangle,  the  sides  of  which  are  m,  ?i,  and  p. 

Let  0  be  the  centre  of  the  circle, 
R  the  radius,  OL  a  perpendicular  to 
MN,  p  =  MN,  and  the  other  sides  as  in- 
dicated in  the  figure. 

Now     NL  =  |,  and  angle  NOL = P. 
.-.  £  =  12  sin  P, 


P 

R  =  suTP  =  2  sin  P 

To  find  an  expression  for  R  in  terms  of  the  three  sides,  sub 
stitute  for  sin  P  its  value 


232  PLANE   SURVEYING. 


2  sin  *P 
whence    fl  = 


£s  -  m)  (is  -  »)  (|s  -p) 
in  which  s  represents  the  sum  of  the  sides  of  the  triangle. 

ADDITIONAL  EXAMPLES. 

1.  Circumscribe  a  circle  about  a  triangle  the  sides  of  which 
are  10,  15,  and  20  chains. 

2.  Find  an  expression  for  the  radius  with  which  to  inscribe 
a  circle  in  a  triangle  the  sides  of  which  are  m,  w,  and^>. 

Ans.     Twice  the  area  of  the  triangle,  divided  by  the  sum  of 
the  sides. 

3.  Describe  a  circle  in  a  triangle  the  sides   of  which   are 
30,  40,  and  50  rods. 

5.  A  circular  walk,  6  feet  wide,  is  to  be  made  inside  of  a 
square  which  contains  \  an  acre  ;  required  the  area  of  the  walk. 

5.  The  area  of  a  square  is  1  acre,  and  a  circular  walk  is 
required  to  be  made  in  it,  touching  each  side  at  a  point,  of  such 
a  width  that  it  will  take  up  \  the  area  of  the  square.     Find  the 
width  of  the  walk  and  the  length  of  its  centre  line. 

6.  The  area  of  a  circular  sector  of  d°  is  m  rods  ;    find  an 
expression  for  the  radius.     If  d  =  60  and  m  =  300,  find  R. 


SECTION  II. 
DIVIDING  LAND. 

A.    TRIANGLES. 


275.    To  divide  a  given  triangle  into  two  parts  in  the  ratio  of 
m :  n  by  a  line  parallel  to  one  side. 


LAYING   OUT   AND   DIVIDING   LAND.  233 

To  solve  the  problem  fully,  and  furnish  a  check  on  the  work, 
requires  the  location  of  the  point  0  or  72,  and  the  length  of 
OR.    Denote  OR  by  x,  OP  by  ?/,  and  by  p  and  7c  the  sides  respec- 
tively opposite  the  angles  P  and  K;  then 
p2 :  x2  =  ra  +  n  :  ra, 


Again,  ft2 :  y2  =  m  +  n  : 

whence 


If  the  triangle  is  to  be  equally  divided,  then  m  =  n,  and  there 
results 

x_p  and       _k   r 

QUERIES.    Is  it  necessary  that  LK  be  known  to  find  either 
PO  or  PR?    Must  LKbe  given  to  find  OR? 

EXAMPLES. 

1 .  Find  a  general  expression  for  the  distance  RK  (last  figure) . 

2.  Show  how  to  divide  the  triangle  LKP  into  four  equiva- 
lent parts  by  lines  parallel  to  the  base. 

276.    To  divide  a  given  triangle  into  two  parts  in  the  ratio 
of  m:  n  by  a  line  from  a  vertex  to  the  opposite  side. 

Let  PO  be  the  line,  x  =  LO,  and  p  as  above.     Then,  since  tri- 
angles having  the  same  altitude  are  to  each  P 
other  as  their  bases,  we  have 

p  :  x  =  m  +  n  :  n ; 

whence  x  =    pn    • 

m  +  n 

EXAMPLES. 

1.    Locate  0  on  the  supposition   that  the  triangle  is  to  be 
divided  into  two  equivalent  parts. 


234  PLANE   SURVEYING. 

2.  Find  where  the  lines  from  P  will  meet  the  base  dividing 
the  triangle  into  three  equivalent  parts. 

3.  The  same  for  any  number  n  parts. 

277.   To  divide  a  given  triangle  into  two  parts  in  the  ratio  of 
m  :  n  by  a  line  through  a  given  point  in  one  of  the  sides. 

Denoting  PL  by  x,  and  the  other  sides  in 
the  usual  manner,  we  have 

m  +  n  :  m  =  Jco  :  px  ; 

-  ,  triko 

—  p-  -----  »_L  whence      x  =  -  • 

p(m  +  n) 

If  the  parts  are  to  be  equivalent,  m  =  ?i,  and  there  results 


EXAMPLE. 

Show  how  the  given  triangle  LKO  may  be  divided  into  three 
equivalent  parts  by  lines  radiating  from  a  given  point  R. 

NOTE.  The  lines  may  or  may  not  fall  on  the  same  side.  Examine  both 
cases. 

278.  The  same  conditions  as  in  the  last  case,  except  the  tri- 
angle is  to  be  isosceles. 

Using  the  same  notation  and  figure  as  in  that  case,  we  have 
the  following  equality  of  ratios  : 

m  +  n  :  m  =  ko  :  x2  ; 

whence  x  =  J™*<L. 

\ra  +  « 

If  the  parts  are  to  be  equivalent,  m  =  ?t,  and  we  have 


EXAMPLE. 


Show  how  to  cut  off  a  given  area,  in  the  form  of  an  isosceles 
triangle,  from  the  corner  of  a  field,  only  the  angle  being  given. 


LAYING    OUT   AND   DIVIDING   LAND.  235 

279.  The  bearings  of  two  sides  of  a  field  being  given,  to  cut  ojff 
a  triangle  having  a  given  area  by  a  line  running  in  a  given  direc- 
tion and  intersecting  the  given  sides. 

a.  Suppose  the  division  line  is  to  make  a  right  angle  with 
either  side.  Let  LO  and  LQ  be 
the  sides,  the  bearings  of  which  are 
known,  and  PR  the  division  line  per- 
pendicular to  LQ.  The  angle  L  be- 
comes known  through  the  bearings  of 
the  sides  which  include  it,  and  there 
follows 

p  tan  L  =  PR  =  L 

But  \pl  =  area  =  A ; 

hence  \p*  tan  L  =  A, 


and  p  = 


tan  L 


b.  Suppose  the  angle  at  R  is  oblique.  Denote  LR  by  or,  and 
LP  by  //,  and  find  from  the  bearings  the  angles  at  P  and  R. 
Then  from  the  two  equations, 

%xy  sin  L  =  A 

x      sin  P 

and  -  =  -- — - 

y      sin  R 

may  be  deduced 

"ilsinZ/  sinP 

QUERY.  Is  it  necessary  that  the  bearings  of  LO  and  LQ  be 
given  if  the  field  is  triangular  and  the  lengths  of  the  sides  given? 

EXAMPLES. 

1.  The  bearing  of  LO  (last  figure)  is  N.  50°  E.,  and  LQ 
S.  82°  E.  It  is  required  to  find  the  lengths  of  LR  and  PR 
perpendicular  thereto,  so  that  3  acres  may  be  contained  in  the 
triangle  PLR. 


236  PLANE   SURVEYING. 

2.  Suppose  LO  =  10,  LQ  =  8,  and  OQ  =  6  chains.     Find 
the  position  and  length  of  the  division  line  PE,  which,  with  an 
angle  PEL  =  84°,  will  cut  off  a  triangle  PEL  containing  1.5 
acres. 

3.  Show  that  if  three  lines  be  drawn  connecting  the  middle 
points  of  the  three  sides  of  a  triangle,  the  four  triangles  thus 
formed  will  be  equal. 

280.   To  divide  in  a  given  ratio  a  given  triangle  by  a  line  pass- 
ing through  a  given  point  within  it. 

Let  OQE  represent  the  given  triangle,  and  P  the  point  with- 
in ;  DL  the  required  division  line, 
and  DEL  :  LDOQ  =  m  :  n. 

The  point  P  may  be  located  by  co- 
ordinates as  PFand  PE,  lines  parallel 
respectively  to  OE  and  QE  ;  or  by  its 
bearing  and  distance  from  one  of  the 
corners,  as  E ;  or  by  perpendicular 
distances  PF',  PE'  from  the  sides. 
The  distances  PF  and  PE  may  be 

calculated  if  the  direction  and  distance  PE  be  known.     Denote 
PF  by  d,  PE  by  b,  DE  by  x,  and  EL  by  y ;  then 

x:y  =  d:y  —  b,         or  xy  =  bx  +  dy ; 

mqo 
and  xy :  qo  =  m  :  m-\-n,  or  xy  =  • 

mqo 
hence  bx-\-dy  =  ——- 


Or,  substituting  the  value  of  y= m(*°    from  equation  above. 

(*»+.»)• 
we  obtain 

,  dmqo  mqo 

ox  -f- — —  i 

(m  +  n)x      m  -f-  n 

whence,  by  reducing  and  completing  the  square,  there  results 
_  mqo  ±  Vm2g2o2  —  4  bdmqo(m  +  n) 


LAYING   OUT   AND   DIVIDING    LAND.  237 

2bmqo 

mqo  ±  Vm2g2o2  —  4  bdmqo  (m  +  n) 

If  the  question  were  to  cut  off  from  a  corner  of  a  tract  of  land 
a  given  area,  by  a  line  passing  through  a  given  point  within, 
we  might  proceed  more  simply,  as  follows  : 

Denote  the  area  to  be  cut  off  by  A,  and  the  other  notation 
as  alove ;  then 

xy  sin  R  =  2  A, 

and  x:y  =  d:y  —  b; 

whence  there  results 


'x  =  A_±  -VA*  -  2  Abd  sin  R 


y 


b  siu  R 
2Ab 


In  each  of  the  two  preceding  problems  there  are  in  general 
two  division  lines,  as  indicated  by  the  double  sign,  fulfilling  the 
conditions  of  the  question.  The  student  will  point  out  when, 
if  ever,  one  of  these  results  will  not  practically  answer  the  first 
case.  Would  either  result  answer  practically  the  second? 
When,  if  ever,  would  the  result  be  imaginary?  Why? 

If  P  were  located  by  its  distance  PR,  and  the  angle  PRL  or 
PRD,  the  lines  PF  and  PE  could  be  calculated,  as  before  re- 
marked, and  the  solution  above  given  made  applicable  ;  or  we 
may  proceed  as  follows  : 

Denote  PR  by  d,  d  sin  PRD  by  6,  d  sin  PRL  by  c,  and  the 
other  notation  as  above  ;  then 


Substituting  the  value  of  y  from  the  first  equation  in  the 
second,  and  reducing,  there  results, 

aa_2Ax_        2cA 

b    '        bs'mR' 


PLANE   SURVEYING. 


whence 


or. 


jin  5 

2A 


As\nE± 


EXAMPLES. 

1  .  Given  the  three  sides  of  a  triangular  tract  of  land  (see 
last  figure),  QR  =  17,  OQ  =  19,  and  OR  =  22  chains,  to  divide 
it  into  two  equivalent  parts  by  a  line  passing  through  a  point  P, 
within  the  field.  PF  and  PE  =  respectively  4  and  9.50 
chains.  The  location  and  length  of  the  division  line  are  re- 
quired. 


2.  It  is  required  to  cut  off  from  the  angle  0,  which  is  60°,  a 
triangular  field  to  contain  10  acres,  by  a  line  DL  passing  through 
a  point  P.     The  distances  PF  and  PE  being  4  and  12  chains 
respectively,  the  location  and   length  of   the  division  line  are 
required. 

3.  Given  the  angle  ORQ=56°  (see  last  figure  but  one), 
PRL  =  2Q°,  and  PR=  12  chains.     It  is  required  to  cut  off  a 


LAYING    OUT   AND   DIVIDING   LAND.  239 

triangle  DRL,  containing  8  acres,  by  a  line  DL  passing  through 
the  point  P.  The  location  and  length  of  the  division  line  are 
required. 

4.  Divide  a  triangular  piece  of  land  into  three  equal  parts 
by  lines  radiating  from  a  point  within. 

SUGGESTION.  The  locus  of  the  vertices  of  all  triangles  having 
the  base  LN  and  one-third  the  area 
of  LMN  is  a  line  parallel  to  LN  and 
at  |  the  height  PM.  Similarly  for 
any  other  side.  Find  point  of  inter- 
section. P 

5.  Apply  the  principle  employed  in  Example  4  to  divide  a 
triangle  into  three  parts,  in  the  ratio  of  1,  2,  and  3,  by  lines 
radiating  from  a  point  within. 

6.  Given  two  sides  of  a  triangle  6  and  8  chains  ;  it  is  required 
to  locate  a  division  line  which  shall  cut  off  from  the  vertex  an 
isosceles  triangle  whose  area  shall  be  to  the  area  of  the  given 
triangle  as  3  :  4. 

7.  Given  the  sides  of  a  triangle  8,  10,  and  12  chains;  it  is 
required  to  divide  it  into  a  triangle  and  a  trapezium,  the  ratio 
of  the  former  to  the  latter  as  2  :  3,  by  a  line  extending  from  the 
middle  of  the  longest  side  to  some  point  on  the  medium  side. 

The  location  of  this  point  and  the  length  of  the  division  line 
are  required. 

8.  Divide  the  triangle  given  in  Example  7  into  three  equiva- 
lent parts   by  lines  radiating  from  the  middle  of   the  longest 
side.     Locate  the  extremities  of  the  division  lines. 

9.  An  angle  QOP  of  a  field  =  42°  30' ;  it  is  required  to  cut 
off  from  some  point  D,  in  the  line  OP,  by  a  line  DL,  making 
an    angle    LDO  =  78°    30',    a    triangle    containing    2    acres. 
Locate  the  division  line,  and  determine  its  length. 

10.  The  sides  of  a  triangle  are  16,  18,  and  24  chains;  it  is 
required  to  divide  it  into  two  parts  in  the  ratio  of  2:3  by  a 
line  perpendicular  to  the  longest  side.     Locate  the  division  line, 
and  determine  its  length. 


240  PLANE    SURVEYING. 

281.    To  divide  a  given  triangle  in  a  given  ratio  by  a  line 
passing  through  a  given  point  without  it. 

Let  ORQ  represent  the  triangle,  P 
the  point  given  by  the  angle  POQ  and 
distance  OP,  DL  the  line  which 
shall  divide  the  triangle,  so  that 

ODL  :  DLRQ  =  m:n. 
Denote  OP  by  6,  OL  by  x,  OD  by  y, 
*    the  angle  DOL  by  0,  the  angle  POD 

by  0',  and  the  -J^—  part  of  the  area  by  A  ;  then 

m  +  n 

^xy  sin  O=  A;  (1) 

also  }by  sin  0'  =  area  POD,  (2) 

and  i  bx  sin  (  0  +  0')  =  area  P0£.  (3) 

)  —  $bysmO'  =  A.  (4) 


Substituting  in  the  last  equation  the  value  of  y  taken  from 
(1)  and  reducing,  there  results, 


or, 
whence 


/ 

a;  sin  0 

2  Ax  2  A  sin  0' 


b  sin  (0  +  0')      sin  0  sin  (0+0') 
A 


b  sin  (0+0') 


x 


2ylsin  0' 


sin  0  sin  (0+0')      62  sin2  (0+0') 

y  may  be  found  by  substituting  the  value  thus  obtained  for 
x,  and  thence  the  length  of  the  division  line  DL. 


EXAMPLES. 


Given,  in  the  triangle  OQR,  OR  =  18.40  chains,  RQ  =  10.20 
chains,   Q0=20.60  chains,   OP  =9. 50  chains,  and  the  angle 


LAYING   OUT   AND   DIVIDING   LAND. 


241 


POQ  =  28°  30',  to  divide  the  triangle  into  two  parts  so  that 
OLD :  DLRQ  =  3:4.  The  position  and  length  of  the  division 
line  DL  are  required. 

B.    QUADRILATERALS. 

TRAPEZOIDS. 

282.  Given  the  parallel  sides  of  a  trapezoid  and  the  perpen- 
dicular distance  between  them,  to  divide  it  by  a  line  parallel  to  these 
sides  into  tivo  parts  having  a  given  ratio. 

Q R 


0     T 


(c) 

Let  OPQR  (Fig.  c)  be  the  trapezoid,  the  sides  OP,  RQ, 
and  the  perpendicular  distance  QT  between  the  bases  being 
given.  It  is  required  to  divide  it  by  a  line  Z)Z,  so  that 
OPLD :  DLRQ  =  m  :  n  ;  that  is,  practically  to  locate  and  de- 
termine the  length  of  the  division  line  DL. 

Denote  the  lower  base  by  6,  the  upper  base  by  &',  the  perpen- 
dicular distance  between  the  bases  by  /*,  the  perpendicular  dis- 
tance between  the  upper  base  and  division  line  by  a;,  the  length 
of  the  division  line  by  y,  and  the  area  OPQR  by  A.  Draw  QV 
parallel  to  RP;  then  the  similar  triangles  give 

OV:DK=  QT-.QF. 


242  PLANE   SURVEYING. 

Or,  OP  -  QR  :  DL  -  QR  =  QT:  QF. 

Or,  substituting  proper  values, 

b  —  b' :  y  —  b'  =  h  :  x ; 
-  6')  h 


whence 

b —  b 

But  the  area  of  DLQR  =  (y  +  &')— =  -^ — 

2      m  +  n 

Representing  for  convenience  the  right-hand  member  of  the 
last  equation  by  A',  we  may  write 

xy  +  b'x  =  2  A', 


and  y  =         -V.  (2) 

Substituting  the  value  of  x  from  (1)  in  (2)  and  reducing, 
there  results 


and 


b  —  b' 
Restoring  the  value  of  A',  we  obtain, 


_  vh  ±  J-2L-  2  Ah  (b  -  b')  +  6'  2h2 
\  m  +  n 


The  student  may  indicate  how  he  would  trace  out  on  the  field 
the  division  line  thus  found. 

283.  If  instead  of  the  perpendicular  distance  there  be  given 
one  of  the  sloping  sides,  as  OQ  (Fig.  c). 

Denote  OQ  by  cZ,  OD  by  x,  and  the  other  notation  as  above. 
Produce  the  sides  until  they  meet  in  some  point  E ;  then 


LAYING   OUT  AND   DIVIDING   LAND.  243 


OPE  i 
DLE:  QRE  =  y2:b12; 

or,  by  division,       OPQR  :  QRE  =  b2  -  b'2  :  6'2, 
and  DLQR:QRE  =  y2-b'2:b'*-, 

whence  OPQR  :  DLQR=  b2  -  b12:  y2-V*. 

By  division          OPLD  :  DL  QR  =  62  -  y2  :  f  -  6'  2  ; 
inserting  values,     m  :  n  =  b2  —  y'2  :  y2  —  b'  2  ; 


whence 


m  -{-n 
The  similar  triangles  OVQ  and  QDKgive 

b-b':y-b'  =  d:d-x. 
_d(b-y): 


\b2n+V2m 


I 
J 


In  Figure  d,  the  unknown  sides  are  symmetrical  with  respect 
to  :i  line  joining  the  centres  of  the  parallel  sides ;  in  Figure  e, 
PR  is  perpendicular  to  the  parallel  sides.  The  student  will 
show  what  modification,  if  anv,  may  be  made  in  the  formulas 
of  the  two  preceding  cases  for  either  of  these. 

EXAMPLES. 

1.  Given  OP=  20  chains,  QR=  15  chains,  QT=  18  chains, 
to  find  the  length  of  the  division  line  DL,  so  that  QRLD  shall 
contain  two-thirds  as  much  land  as  OPLD. 

2.  In  Figure  d,  whose  sides  are  equally  inclined  to  the  bases, 
OP=  24  chains,  QR  =10  chains,  and  the  perpendicular  distance 
QT=20  chains;  it  is  required  to  locate  the  extremities  of  the 
division  line  DL,  and  determine  its  length,   so  that  it  shall 
divide  the  tract  into  two  equivalent  parts. 


244  PLANE   SURVEYING. 

3.  In  Figure  e,  suppose  QR  :  OP :  PR  =  3  :  4  :  5,  and  that  the 
area=  1750  rods  ;  locate  and  find  the  length  of  the  division  line 
DL  that  shall  divide  the  tract,  making  OPLD :  QRLD  =3:4. 

284.  To  divide  a  given  trapezoid  into  two  parts  having  a 
given  ratio,  by  a  line  intersecting  the  parallel  sides. 

Let  OPQR  represent  the  trapezoid,  and  let  it  be  required  to 
divide  it  into  two   equal   parts.     It   is 
evident  if   the  bases   be  bisected,  and 
,      i  /         \          a  line,  as  DL,  be  drawn  connecting  the 
™~^         points  of  division,  it  will  be  the  division 
line  required. 

Similarly,  if  the  ratio  is  m  :  n  ;  denote 

OP  by  b,  and  RQ  by  b' ;  then  take  OL  =  _J^_  RD  =-^-, 
and  join  DL  for  the  line  required. 

The  student  will  give  the  reason. 

If  the  division  line  is  to  pass  through  a  given  point  D',  ob- 
tain DL  as  above  directed,  then  measure  from  D  to  D',  and 
lay  off  this  distance  from  L  to  L'.  Join  D'L'  for  the  division 
line  required.  Why? 

To  divide  a  trapezoid  by  a  line  perpendicular  to  the  bases, 
or  parallel  to  one  of  the  non-parallel  sides,  divide  the  line  join- 
ing the  middle  points  of  the  non-pnrallel  sides  into  two  parts  in 
the  given  ratio,  and  through  the  point  of  division  run  the  re- 
quired line.  If  m :  n  is  the  ratio,  and  the  bases  b  and  &',  the 

distance  TKm  the  last  figure  =  w(&  +  6>). 
2  (m  +  n) 
The  student  will  give  the  reason. 


EXAMPLES. 

1.  Divide  a  given  trapezoid  into  three  equivalent  parts  by 
lines  intersecting  the  parallel  sides. 

2.  Divide  a  given  trapezoid  into  three  parts  in  the  ratio  of 
m :  n  :p,  by  lines  intersecting  the  parallel  sides. 


LAYING    OUT   AND   DIVIDING    LAND.  245 

3.  The  bases  of  a  trapezoid  are,  OP  =20  chains,  and  QR  = 
15  chains.     It  is  required  to  divide  it  into  two  parts  in  the  ratio 
of  2  :  3.     OL'  =  8.50  chains  ;  locate  D'. 

4.  Show  that  /,  being  the  centre  of  the  line  connecting  the 
middle  of  the  bases  of  a  trapezoid,  is  the  point  through  which, 
if  any  straight  line  be  drawn  meeting  the  parallel  sides,  it  will 
divide  the  trapezoid  into  two  equivalent  parts. 

285.  Given  one  side  and  the  adjacent  angles  of  a  tract  of  land, 
to  cut  off  a  trapezoid  of  a  gicen  area  by  a  line  parallel  to  the 
given  side. 

Let  PO  be  the  given  base,  P  and  0  the  known  angles  indi- 
cating  the   direction   of  the  p 
sides  PQ  and  OR.     Denote 
the   area  OPLD,  to  be  cut 
off   by   A ;    the   given    side             ..---" 
OP  by  s,  PD  by  y,  OL  by     F"": 
ce,  DL  by  z,  and  suppose 

(0-f-P)<180°. 

Produce  OR  and  PQ  until  they  meet  in  F. 
Then  area  OPV  -  area  LDV  =  A  ; 

s2  sin  0  sin  P     z2  sin  0  sin  P 


2 
whence  z  =  \('s2  -- 

M  sin  0  sin  P 

When  (0  +  P)  >  180°,  the  produced  lines  meet  in  a  point  on 
the  other  side  of  OP,  the  sin  (0  +  P)  is  also  negative,  and 
therefore  the  fraction  under  the  radical  becomes  positive. 
Draw  LT  parallel  to  VP;  then  in  the  triangle  LOT,  by  sine  pro- 
portion, sin  L  (=  sin  F)  :  sin  T  (  =  sin  P)  =  s  —  z  :  x  ; 

whence  X=S 


sinF 

c,.    .,  _,  (s  —  z)  sin  0 

Similarly,  y  =  i  —         -- 


246 


PLANE   SURVEYING. 


REMARK.  When  great  accuracy  is  not  required,  and  espe- 
cially if  the  tract  is  small  and  the  sides  nearly  parallel,  an 
approximate  perpendicular  distance  between  the  bases  OP  and 
DL  may  be  obtained  by  dividing  the  area  to  be  cut  off  by  the 
given  side  OP;  then  measure  the  perpendicular  and  a  line 
through  its  extremity  parallel  to  the  base  for  an  approximate 
division  line.  Calculate  the  area  thus  cut  off,  divide  the  differ- 
ence between  it  and  the  required  area  by  the  approximate  divis- 
ion line  for  a  new  perpendicular,  and  thence  obtain  more  nearly 
the  division  line  sought. 

EXAMPLES. 

1.  Deduce  an  expression  for  DL  by  another  method. 

2.  Show  by  other  methods  how  OL  or  PD  may  be  determined. 

3.  Given  OP,  N.  16°  30'  W.,  8.40  chains  ;  PQ,  S.  62°  15'  W ; 
and  OR,  S.  82°  W.,  to  cut  off  a  trapezoid  containing  4  acres, 
by  a  line  DL  parallel  to  OP.     The  position  and  length  of  the 
division  line  are  required. 

4.  Given  a  side  of  a  tract  of  land  20  chains,  and  the  adja- 
cent angles  105°  and  130°,  to  cut  off  36  acres  by  a  line  parallel 
to  the  given  side.      Required  the  position  and  length  of   the 
division  line. 

TRAPEZIUMS. 

286.  Given  the  area  of  a  trapezium,  one  of  its  sides  and  adja- 
cent angles,  to  divide  it  by  a  line  parallel  to  the  given  side  into  two 
parts  having  the  ratio  m  :  n. 


Produce  the  sides  PQ  and  OR  to  meet  in  V.     Let  OP=  s, 
OL  =  x,  PD  =  y,  DL=  z. 


LAYING   OUT   AND   DIVIDING   LAND.  247 

Calculate  the  area  of 

' 


•2  siuF 
then  A'-A" 


j  ,,     »         ,       22  sin  0  sin  P        ,  A,       Au\ 
and  the  formula     -         -  =  2(  A1  —  A) 
sinF 


gives  )2sinF(.i'-^. 

\      sin  0  sin  P 

Having  found  3,  x  and  y  ma^y  be  deduced  as  in  the  foregoing 
case. 

(s  —  z)  sin  P 

sinF 

(s-z)sin  0 
sinF 

REMARK.    This  problem  may  be  solved  by  Article  285,  taking 
for  the  given  area  to  be  cut  off  — — — ' 


EXAMPLE. 
The  boundaries  of  a  trapezium  are  as  follows : 

(1)  N.      2°  E.     8.00  chains; 

(2)  N.  58£°  E.   13.85      " 

(3)  S.  3H°  E.   14.80       k' 

(4)  S.  82^°  W.  20.00       " 

It  is  required  to  divide  it  into  two  equivalent  parts  by  a  line 
parallel  to  the  third  side.  Locate  it,  and  determine  its  length. 

287.  Given  the  bearings  of  three  adjacent  sides  of  a  tract  of 
land  and  the  length  of  the  middle  one.  to  cut  off  «  trapezium 
having  a  given  Jtrat,  by  a  line  running  in  a  given  direction. 


248  PLANE   SURVEYING. 

Produce  the  sides  PQ  and  OR  till  they  meet  at  F.  As 
before,  denote  OP  by  s,  OL  by  ar,  PZ)  by  ?/,  and  .LZ>  by  z.  Ob- 
tain the  angles  from  the  bearings,  calculate  the  area  of 

PQY=  A'  =  ** sin  °  sin  f \ 
2  sin  F 

and  find  area      DL  F=  ^'  -  A. 


R 

L 

0 


Whence  the  division  line  DL  =  z  may  be  found  from  the 
formula  22sinZ>siu£       Al 

—  -ZL    "~~  •*}.) 


2siuF 

z  = 


sin  D  sin  Zr 
By  the  sine  proportion 


sin  F 
and  FL  = 


whence  VO-VL=LO  =  x  =n-sn 

sin  F 
and  _  ssin  0  —  zsin  L 

sin  F 

REMARK.  If  (0  +  P)  >  180°,  ^f-^4  in  the  equation  for  z 
will  become  A1  -f-  .4,  and  in  the  formulas  for  x  and  y  the  signs 
in  the  numerators  will  be  interchanged,  or 

—  z  s*a  -^  ~ 
sin  F 

and  y  =  zsin^~'ss 

sin  F 


LAYING   OUT   AND   DIVIDING   LAND.  249 

EXAMPLE. 

Given  LO,  S.  76°  E. ;  OP,  N.  8°  W.  12.40  chains  ;  PD,  S.  72° 
W. ;  it  is  required  to  cut  off  7  acres  bv  aline  hearing  N.  23°  W. 
The  length  of  the  division  line  and  the  distances  OL  and  DP 
are  to  be  computed. 

288.  Given  a  trapezium,  to  divide  it  into  two  parts  having  a  given 
ratio,  by  a  line  extending  from  a  given  point  in  one  of  the  sides. 

Let  OPQR  represent  the  trapezium  the  area  of  which  is  A, 
m  and  n  the  given  ratio.  Prolong  the  sides  PQ  and  OR  till 
they  meet  in  F.  Let  OR  =  v,  the  division  line  DL  =  z,  RL  the 
given  distance  to  the  point  L=  d,  and  QD  =  y.  Calculate  the 
area  of  QRV=A',  and  add  it  to — - — A,  thereby  obtaining 
areaofZ)£F.  p 

Find  by  the  sine  proportion  VR, 
and  add  it  to  RL,  thus  obtaining 
VL. 

Putting  VD  =  x,  and  VL  =  i>,  .-• 

V='2( — - — A+A\ 


Whence  x  =  VD  may  be  found. 

Finally,  with  the  two  sides  VD  and  VL  and  the  included 
angle  V,  compute  the  angle  L,  and  the  direction  and  length  of 
the  division  line  DL  ;  y  may  be  calculated  by  a  preceding  method 
to  check  the  work. 

EXAMPLES. 

1.  Given  in  a  trapezium  MNOP  (no  figure)  : 
MN,  13.00  chains ; 
NO,      7.30      " 
OP,    10.40      " 
PM,   11.10      " 
and  diagonal  PN,   13.70      " 

It  is  required  to  divide  it  into  two  equivalent  parts  by  a  line 
running  from  a  point  in  the  side  MN,  6  chains  from  M.  Find 


250  PLANE   SURVEYING. 

the  length  of  the  division  line  and  locate  the  other  extremity 
of  it. 

2.  Divide  the  tract  described  in  Example  1  into  two  parts,  in 
the  ratio  of  3  :  4,  by  a  line  DL  running  from  some  point  in  MN. 
and  falling  perpendicularly  upon  PO.  The  part  PMDL  is  to  be 
the  greater.  Locate  the  line  required,  and  determine  its  length. 

289.  Given  a  trapezium,  to  divide  it  into  two  parts  having  a 
given  ratio,  by  a  line  passing  through  a  given  point  within  the 
tract. 

Let  OPQR  represent  the  given  trapezium  T,  the  point  within 
it,  given  by  its  bearing  and  dis- 
tance from  some  angle,  as  R. 
Produce  the  sides  OR  and  PQ  to 
meet  in  V.  Denote  the  ratio  by 
ra  and  n,  the  area  OPQR  by  A, 
QR  by  v,  DL  by  z,  VL  by  «,  and 
VI)  by  y.  Find  by  the  sine  pro- 
portion 

yR  _  vsinQ     VQ  _  v  sin  R 
sin  V"  sin  F' 


and  thence  the  area  VQR  =  A'.  Then  in  the  triangle  VRT, 
having  two  sides  and  the  included  angle,  compute  FT,  which 
call  b,  and  the  angle  TVR  =  a.  Putting  F—  a  =  ft,  and 
— - —  A  +  A'  =  A",  the  following  equations  may  be  written : 

xysinV=2A",  (1) 

and  bx  sin  a  +  by  sin  ft=  2  A".  (2) 

Substituting  in   (2)  the  value  of  y  from  (1),  and  reducing, 
there  results, 

x  =     A"    ±     I    A"2         2  A"  sin  ft  ^ 
b  sin  a.       \  b~  sin2  a       sin  a  sin  F 


*rtEL=x-VR=-^-±  J    A"2     _*A'*infl_vunQ 
b  sin  a        \  b2  sin2  a       sin  a  sin  F       sin  F 


LAYING   OUT   AND   DIVIDING   LAND.  251 

EXAMPLE. 
Given  the  boundaries  of  a  trapezium  as  follows : 

(1)  N.  16J°  W.  24. 63  chains; 

(2)  S.     79°  W.  27.00      " 

(3)  S.       i°W.  34.28      » 

(4)  N.     65°  E.  37.20      " 

To  divide  it  into  two  equivalent  parts  by  a  line  extending 
from  the  first  to  the  third  side,  and  passing  through  a  point 
20  chains  distant  from  the  first  and  second  corners.  Locate 
the  line  and  find  its  length. 

290.  Given  a  trapezium,  to  divide  it  into  two  parts  having  a 
given  ratio,  by  a  line  passing  through  a  given  point  without  the 
tract. 

P 


Let  OQRT  represent  the  trapezium  given  by  the  bearings 
and  distances  of  its  sides,  P  the  point  without,  located  by  its 
bearing  and  distance  from  T,  the  ratio  m  :  n.  Extend  the  sides 
RT  and  QO  until  they  meet  in  V.  Then  the  problem  may  be 
solved  in  a  similar  manner  to  that  in  Article  281. 

291.  Given  a  trapezium,  to  divide  it  into  four  equivalent 
parts,  by  two  lines  intersecting  opposite  sides,  one  of  the  division 
lines  being  parallel  to  one  of  the  given  sides  of  the  tract. 

Let  ABCD  represent  the  given  trapezium,  FE  the  division 
line  parallel  to  DC,  and  GH  the  other  division  line.  It  is  re- 


252 


PLANE   SURVEYING. 


quired  to  locate  both  division  lines.  Prolong  the  sides  AD  and 
BCto  meet  in  P;  also  DC  and  AB  to  Q.  Find  AE  and  EF 
by  methods  already  given. 

Now,  any  line  cutting  the  parallel  sides  of  a  trapezoid  and 
dividing  it  into  two  equivalent  parts  must  pass  through  a  point 
0  (the  middle  of  the  middle  line  between  the  bases).  See 
Article  284.  Hence  MO  becomes  known  =  1  (CD  +  EF),  and 


P  Br- 


aise MC=\FC.  In  the  triangle  OMC,  compute  the  angle 
MCO  and  the  line  0(7;  add  Z  MCO  to  Z  MCQ,  and  having 
previously  calculated  QC,  find  in  the  triangle  QCO  the  angle 
CQO  and  the  side  QO.  Subtract  Z  CQO  from  Z  CQB  and 
obtain  Z  OQ-B.  Then  putting  the  side  QO  =  a,  QH=x,  and 
Q(?  =  y,  we  may  write  the  following  equations : 

ay  sin  HQG  =  2  area  HQG, 

ax  sin  CQO  +  ay  sin  OQG  =  2  area  #Q#. 

From  these  equations  obtain  iy.  Subtract  it  from  AQ,  found 
by  sine  proportion,  and  the  distance  from  the  corner  A  to  the 
extremity  of  the  division  line  GH  at  G  will  be  the  result. 

Then  in  the  triangle  QGH  find  QH',  whence  the  length  and 
bearing  of  GH  may  be  computed. 


LAYING   OUT  AND  DIVIDING   LAND.  253 

EXAMPLE. 

It  is  required  to  divide  the  farm  described  in  288  (Example  1 ) 
into  four  equivalent  parts  by  two  lines  intersecting  opposite 
sides  ;  one  of  the  division  lines  is  to  be  parallel  to  the  first  side. 
Locate  the  division  lines,  and  determine  their  lengths. 

C.    POLYGONS. 

292.  Given  a  polygon,  to  divide  it  into  two  parts  having  a 
given  ratio,  or  to  cut  off  a  given  area,  by  a  line  through  a  given 
point. 

Let  OPQRTV 'represent  the  polygon  given  by  its  bear- 
ings and  distances,  or  angles  and  sides,  0 v 

and  suppose  the  line  be  required  to 
run  from  P,  either  an  angle  or  any 
given  point  in  a  side.  Calculate  the 

area  of  the  polygon ,  and  take  the  — — — 

m  +  n 

part  of  it  as  the  area  to  be  cut  off  to 
the  right  of  the  line  extending  from  P.  Q  -B 

Run  a  trial  line*  from  P  as  PT,  calculate  the  area  of  PQET, 
and  determine  whether  the  area  thus  cut  off  is  too  small  or  too 
large,  and  how  much.  Suppose  it  is  too  small ;  then  the  extrem- 
ity T  of  the  division  line  PT  must  be  moved  towards  V  to 
some  point  T1.  To  find  this  point,  denote  TT1  by  x,  the  angle 
TTP  by  T,  the  distance  PTby  b,  and  the  area  of  the  triangle 
PTT  by  a ;  then 

£  bx  sin  T  =  a, 

from  which  we  find        x  =  — — — 
b  sin  T 

*  The  bearing  and  distance  of  PT  may  be  calculated  from  the  data 
given  —  without  a  trial  line — as  in  supplying  omissions.  If,  however, 
this  is  done,  the  surveyor  should  not  omit  to  measure  the  division  line  to 
verify  his  work.  In  fact,  it  is  the  best  practice,  no  matter  what  method  is 
adopted  to  obtain  the  division  line,  to  always  test  the  computation  by 
measurement. 


254  PLANE   SURVEYING. 

This  distance  measured  from  T  to  T  will  locate  T',  a 
point  which  connected  with  P  will  give  the  division  line 
sought. 

293.  Given  a  polygon,  to  divide  it  into  two  parts  in  a  given 
ratio,  or  to  cut  off  a  given  area,  by  a  line  through  a  given  point 
within  the  tract. 

Let  the  marginal  figure  represent  the  tract,  Pthe  given  point. 
If  the  area  to  be  cut  off  is  not  di- 
rectly given,  calculate  the  contents  of 
the  tract,  and  then  by  the  ratio  deter- 
mine the  quantity  to  be  cut  off,  and 
denote  it  by  A.  Run  a  trial  line  TT 
through  P,  dividing  the  polygon  as 
nearly  as  may  be  judged  in  the  re- 
quired manner.  Measure  TP=  b,  PT' 
=  c,  and  the  angles  T  and  T'.  Calcu- 
late the  area  of  either  part  of  the  polygon,  and  thus  ascertain 
whether  T  should  approach  or  recede  from  0.  Suppose  the 
area  TNMVT'  is  calculated  and  found  too  small  by  a  quan- 
tity a,  and  that  DL  represents  the  division  line.  Put  DP=  x, 
PL  =  y,  the  angle  PTD  =  T,  PT'L  =  T',  and  the  angle  at  the 
point  P  =  P,  which  is  required,  since  that  will  indicate  the  di- 
rection of  the  division  line. 

Then  £  cy  sin  P  —  %  bx  sin  P  =  a  (1) 

•=-^^W  (2) 


(3) 


sin  (T'  +  P) 


Substituting  the  values  of  x  and  y  from  (2)  and  (3)  in  (1), 
there  results 

c2  sin  T'  sin  P     b2  sin  T  sin  P 


in(7"  +  /')        sin(T+P) 


=  2  a.  (4) 


LAYING   OUT   AND   DIVIDING   LAND.  256 

Expanding  the  denominators,  dividing  each  fraction,  numer- 
ator and  denominator,  by  its  numerator,  and  writing  for  —t 
the  cot,  there  results  »sin 

fS  7,2 

.  (5) 


cot  P  +  cot  T'      cot  P  +  cot  T 

Putting  cotP  =  ^>,  cot!T=£,  and  cot7"  =  £',  we  may  write 
more  simply  : 


/  c*-l>*\       tc*  —  W 

(t  +  f  -  -^-)  =  —^—  ~  tf  ; 

whence  - 


2a 
Restoring  values,  we  have 


cotP=  — -f  cot  T+cot  T' 


2a    ) 


_cot  T  cot  r+ 


\  2  a  4\  2a 

The  problem  may  be  simplified  when  it  is  practicable  to  run 
the  trial  line  at  right  angles  to  one  of  the  sides  of  the  polygon. 
In  the  tract  given,  suppose  TT'  to  be  run  perpendicularly  to 
R  F;  then  cot  T'  =  0,  and  Equation  (5)  may  be  written 

*          v 


cot  P      cot  P  +  cot  T 


& 


4V  2o 


256 


PLANE   SURVEYING. 


294.    Given  a  polygon,  to  cut  off  a  given  area  by  a  line  passing 
through  a  given  point  without  the  tract. 

Let  the  marginal  figure  represent  the  case. 
P    h       y  M  As  in  the  preceding  article,  run 

a  trial  line  PT'  from  P,  and  sup- 
pose it  is  made  perpendicular  to 
v  RV.  Calculate,  as  before,  the  con- 
tent  of  TNMVT',  and  ascertain 
the  amount  to  be  added  to  make 
the  required  area.  Denote,  as 
before,  this  area  by  a,  PT=b, 
the  angles  at  P,  T,  etc.,  by  P,  T, 


R 

PT'  =  c,  PD  =  x,  PL  =  y 
etc.,  and  DL  the  division  line  ;  then 

^  cy  sin  P  —  |  bx  sin  P = a, 
6  sin  T 


(1) 
(*) 


Substituting  the  values  of  x  and  y  from  (2)  and  (3)  in  (1), 
and  reducing  as  in  the  preceding  problem,  there  results 

cot,P=-— 


The  student  may  verify  the  value  found. 

295.    Given  a  polygon,  to  divide  it  into  three  parts  having  a 
given  ratio,  by  lines  radiating  from  a  given  point. 

a.  Let  the  figure  represent  the  polygon, 
and  suppose  the  point  is  in  one  side  at 
P.  Calculate  the  area  of  the  whole  tract 
and  ascertain  how  much  each  division  is 
to  contain  ;  then,  by  Article  292,  cut  off 
the  required  areas  PVTSL  and  POQD, 
and  the  problem,  is  solved. 


LAYING   OUT    AND   DIVIDING   LAND. 


257 


6.  If  the  point  is  within  the  tract,*  cut  off,  by  Article  293, 
one  required  portion  DVTSKD  by  a  line  DK  through  P,  and 
by  the  preceding  article  divide  the  remainder  by  the  line  PL  as 
required.  If  PL  cuts  off  a  quadrilateral  on  either  side,  Article 
288  may  be  used. 


c.  If  the  point  is  without,*  proceed,  as  in  Article  294,  to  cut 
off  the  required  portion  KVTSDK  and  HOQLH;  the  remain- 
der HLRDKHvt\\\  be  the  third  portion. 

It  is  evident  that  this  principle  may  be  extended  to  any 
number  of  parts. 

296.  To  cut  off  from  a  given  polygon  a  given  area  by  a  line 
running  in  a  given  direction. 

Let  the  figure  represent  a  tract  which  it  is  required  to  divide 
into  two  equivalent  parts  by  a  line  DL  parallel  to  RS. 

Join  QT,  calculate  its  length 
and  bearing,  and  also  the  content 
of  QRST.  Subtract  said  content 
from  one-half  the  area  of  the  whole 
tract,  thereby  obtaining  the  area 
DLTQ.  Then,  by  Article  287,  the 
length  and  position  of  the  division 
line  may  be  determined.  It  is 
evident  that  this  principle  may  be  extended  to  any  number  of 
subdivisions. 


*  Calculate  the  area,  and   ascertain,   by   the   ratio,  how  much   eacb 
division  is  to  contain. 


258 


PLANE   SURVEYING. 


EXAMPLES. 

1.  The  student  may  indicate  how  he  would  divide  DQRSTL 
into  two  equivalent  parts  by  a  line  perpendicular  to  DL. 

2.  Show  how  to  divide  DPONMVL  into  two  equivalent  pails 
by  a  line  extending  from  the  middle  of  DL. 

3.  Divide  the  farm  described  in  Article  234,  Example  4,  into 
two  equivalent  parts  by  a  line  running  due  east. 

297.  From  a  tract  of  land  of  which  one  or  more  of  the  boun- 
dary lines  is  irregular,  to  cut  off  a  given  area. 

L nQ       Let     OPQRT    represent     the    tract 

which  it  is  required  to  divide  into  two 
equal  pails  by  a  line  DL  parallel  to  PQ. 
Survey  the  land,  taking  offsets  along 
RO,  and  calculate  the  area.  Then 
the  problem  may  be  solved  by  Article 
285. 

298.  To   Straighten  Boundary  Lines.     It  is  sometimes  re- 
quired to  substitute  a  straight  line  for  an  irregular  or  crooked 
one  between  farms,  and  to  leave  the  same  quantitv  of  land  as 
before  in  each  tract.     Let  ORQ  be  the  line  which  it  is  required 


to  straighten  by  a  line  extend- 
ing from   0,   the  bearings  and 


distances  OR,  RQ,  and  the  bear- 
ing of  QT  being  known.  Run  a 
trial  line  OP,  noting  the  dis- 
tances RK,  OK,  KP,  and  PQ, 
and  calculate  the  areas  of  the 
V  triangle  ROK  and  PQK.  If  it 
happens  that  the  triangle  ROK  is  equivalent  to  PQK,  then  OP 
will  represent  the  line  sought.  If,  as  is  generally  the  case, 
their  areas  are  not  equal,  take  the  difference,  and  suppose  in 
this  case  PQK  the  less.  The  problem,  then,  is  simply  this  : 
Given  one  side,  OP,  of  a  triangle,  and  the  direction  of  another, 


LAYING   OFT   AND   DIVIDING    LAND.  259 

PT,  to  cut  off  a  given  area  by  a  line  OP*,  to  find  the  distance 
PP.  The  solution  is  given  in  Article  257. 

Otherwise,  with  the  given  bearings  and  distances  calculate 
the  area  of  the  triangle  ORQ  and  the  length  and  bearing  of  the 
closing  line  OQ.  Then,  as  before,  having  one  side  of  a  triangle, 
the  direction  of  another,  and  the  area,  find  QP1  and  the  bearing 
and  distance  of  OP.  The  work  should  be  verified  by  actual 
measurement  of  angle  and  distance. 

EXAMPLE. 

Given  OR,  N,  59°  30'  E.  10.60  chains;  KQ,  S.  70°  15'  E. 
19.32  chains ;  QT,  N.  12°  W.,  to  find  QP'  and  the  bearing  and 
distance  of  a  line  OP'  which  will  straighten  the  boundary. 

MISCELLANEOUS  EXAMPLES. 

1.  It  is  required  to  lay  out  a  lot  to  contain  one  acre,  and 
having  an  equal  frontage  on  two  streets  which  intersect  at  an 
angle  of  84°  40'.     Locate  the  corners  of  the  property. 

2.  From  a  square    tract  of    land   OPQR,   which   originally 
contained     160     acres,     the     southwest 

quarter  was  sold.  It  is  required  to  find 
the  uniform  width  of  a  strip  MNL  VTS 
which  shall  contain  40  acres.  How  many 
rods  of  fencing  will  the  tract  require? 

3.  A  rectangular  tract  of  land  16.20 
chains  long,  and  8. 60  chains  wide,  valued 
at  $200  per  acre,  is  to  be  divided  among 

three  persons  so  that  the  first  shall  have  $1,000  worth  of  it ;  the 
second,  $900  ;  and  the  third,  the  remainder.  Locate  the  points 
of  division  on  the  long  side. 

4.  The  bearings  of  two  sides  of  a  triangle  are  OM,  N  60°  E., 
and  ON,  S.  40°  E.     It  is  required  to  cut  off  from  the  corner  0 
an  isosceles  triangle  containing  16  acres.     Locate  and  find  the 
length  of  the  division  line. 


260  PLANE   SURVEYING. 

5.  There  is  a  farm  in  the  form  of  a  trapezium  the  area  of 
which  is  given  as  87.78  acres.     The  description  of  its  boun- 
daries is  very  much  effaced  ;  all  that  is  legible  is  as  follows  : 

Beginning  at  the  northwest  corner,  thence  (1)  S.  76°  E. 
(distance  effaced)  ;  (2)  S.  10°  E.,  distance  25  chains;  (3)  S. 
62°  W.  (distance  effaced)  ;  (4)  N.  6°  W.  (distance  effaced). 

It  is  required  to  perfect  the  description. 

SUGGESTION.  Prolong  the  second  and  fourth  sides  until  they 
meet,  and  calculate  the  area  of  the  triangle  exterior  to  the  tract. 
Add  it  to  the  given  area,  whence  the  length  of  the  first  side 
may  be  readily  computed  ;  the  second  and  fourth  sides  may  be 
found  easily  by  either  of  two  methods. 

6.  Required  the  length  of  a  chord  which  will  cut  off  one-third 
part  of  a  circle  whose  radius  is  100  feet. 

SUGGESTION.     Let  20  denote  the  central  angle,  and   r  the 

radius,  for  convenience.     Then ^--—  sin 20  =— •    Whence 

180       2  3 

0  may  be  obtained,  and  hence  the  chord.     The  angle  will  be 
the  same,  no  matter  what  the  radius  may  be. 

7.  A  trapezoidal  field,  the  two  parallel  sides  of  which  are  16 
and  10  chains,  and  the  perpendicular  distance  between  them,  12 
chains,  is  to  be  divided  into  two  equivalent   parts  b3*  a  line 
parallel  to  the  given  sides.      It  is  required  to  determine  the 
length  of  the  division  line  and  locate  its  extremities,  the  sides 
being  equally  inclined  to  the  bases. 

8.  Given  the  sides  of  a  triangle  OR,  280  yards ;  RQ,  200 
yards  ;  OQ,  300  yards  ;  the  distance  from  0  to  a  point  P  outside 
the  tract,  220  yards  ;  and  the  angle  POQ,  20°.     It  is  required 
to  run  the  centre  line  of  a  straight  road  through  P  and  across 
the  field,  so  as  to  divide  the  tract  into  two  equal  parts.     Locate 
the  points  where  the  road  will  cross  the  triangle. 

9.  Given  the  sides  of  an  irregular  pentagon,  and  the  per- 
pendicular distance  to  each  from  a  point  within.     Show  how  to 
divide  the  tracts  into  their  equivalent  parts.     Also  into  three 
parts,  having  the  ratio  m  :  n  :p. 


LAYING   OUT   AND   DIVIDING    LAND.  261 

10.  Given  in  a  trapezoid  MNOP  (no  figure),  PM=  38.50 
chains ;  MN,  one  of  the  parallel  sides,  64.80  chains ;  NO,  41 
chains  ;  the  angle  M,  85°  30f ;  and  N,  75°  40'.    It  is  required  to 
divide  the  tract  into  two  parts  in  the  ratio  of  2  :  3,  by  a  line  DL 
parallel  to  the  parallel  sides.     The  part  MNDL  is  to  be  the 
greater.     Find  the  length  and  location  of  the  division  line. 

11.  Given  one  side  of  a  triangular  field,  120  yards  ;  the  angle 
opposite,   20°;    and  the  ratio  of   the  other  two  sides,   7:10. 
Find  the  area. 

12.  Show  that  the  area  of  a  trapezium  is  equal  to  one-half 
the  product  of  its"  diagonals,  by  the  sine  of  the  angle  of  their 
intersection. 

13.  From  a  point  within  a  triangular  field,  the  sides  of  which 
were  equal,  I  measured  the  distances  to  the  three  angles,  and 
found  them  12.5,   10,  and  7.5   chains  respectively;    required 
the  area. 

Ana.     12  A.  1  R.  23  P. 

The  above  problem  is  given  in  Guinmere's  Surveying,  and  by 
some  surveyors  it  is  considered  difficult.  The  following  is  an 
outline  of  a  solution  ;  the  student  will  supply  what  is  wanting  : 

With  the  given  distances  form  the  triangle  ABC.  On  AB  de- 
scribe an  equilateral  triangle  ABD  ;  join 
CD  by  a  right  line,  and  on  it  describe 
an  equilateral  triangle  CDE.  CDE  is 
the  triangle  in  question,  and  B  the 
point  within.  For  BC  and  BD  are  evi- 
dently two  of  the  measured  distances, 
and  BE,  it  will  be  perceived,  is  the  D 

other,  through  the  similarity  and  equality  of  the  triangles  ADC 
and  BDE.  To  find  the  area  of  CDE,  compute  the  angle  BAG, 
whence  the  angle  CAD  becomes  known  ;  now  with  the  two 
sides  AC,  AD,  and  the  included  angle  CAD,  CD  is  easily 
determined,  and  hence  the  required  area  of  the  triangle  CDE. 


CHAPTER  V. 

PLANE-TABLE   SUEVEYING. 

299.  The  Plane-Table,  as  its  name  indicates,  is  a  table  or 
board   which,  being   covered  with  paper,  and   having   certain 
appliances  for  levelling  and  sighting,  enables  the  surveyor  to 
determine  points  and  lines,  and  to  delineate  them  on  the  paper 
in  their  relative  position. 

It  is  used  in  "filling  in"  the  details  of  topographical  work, 
and  generally  for  the  location  of  points  where  great  accuracy 
is  not  required,  on  account  of  the  rapidity  with  which  surveys 
by  it  may  be  effected. 

300.  The    Board,   which   is   rectangular    in   shape,    usually 
24  by  30  inches,  is  made  of  pieces  of  well-seasoned  wood  joined 
advantageously  together  to  prevent  warping,  and  is  furnished 
with  rollers  or  clamps,  by  means  of  which  the  paper  is  kept 
securely  stretched  upon  it. 

301.  The  Plumbing- Arm,  which  is  pointed  at  one  end,  and 
from  the  other  a  plummet  is  suspended,  is  used  to  determine 
the  point  on  the  ground  immediately  under  its  representative 
on  the  board,  or  vice  versa.    The  lower  part  of  it  moves  upon  an 
axis  which  has  an  index  at  its  extremity,  by  means  of  which  it 
may  be  ascertained  when  the  bob  and  point  upon  the  table  are 
in  the  same  vertical  line. 

302.  The  Tripod  and  its  Head  are  similar  to  those  of  the 
ordinary  transit,  though  heavier. 

A  metallic  plate,  screwed  fast  to  the  table  and  having  a  solid 
conical  spindle  projecting  from  its  centre,  affords  the  means  of 
attaching  the  head  to  the  table. 


PLANE-TABLE, 
As  MADE  BY  HELLER  &  BRIGHTLY,  PHILADELPHIA,  PA. 


CHAPTER   V. 

PLANE-TABLE  SURVEYING. 

299.  The  Plane-Table,  as  its  name  indicates,  is  a  table  or 
board   which,  being   covered  with   paper,  and    having   certain 
appliances  for  levelling  and  sighting,  enables  the  surveyor  to 
determine  points  and  lines,  and  to  delineate  them  on  the  paper 
in  their  relative  position. 

It  is  used  in  "filling  in"  the  details  of  topographical  work, 
and  generally  for  the  location  of  points  where  great  accuracy 
is  not  required,  on  account  of  the  rapidity  with  which  surveys 
by  it  may  be  effected. 

300.  The    Board,   which   is   rectangular   in   shape,    usually 
24  by  30  inches,  is  made  of  pieces  of  well-seasoned  wood  joined 
advantageously  together  to  prevent  warping,  and  is  furnished 
with  rollers  or  clamps,  by  means  of  which  the  paper  is  kept 
securely  stretched  upon  it. 

301.  The  Plumbing- Arm,  which  is  pointed  at  one  end,  and 
from  the  other  a  plummet  is  suspended,  is  used  to  determine 
the  point  on  the  ground  immediately  under  its  representative 
on  the  board,  or  vice  versa.    The  lower  part  of  it  moves  upon  an 
axis  which  has  an  index  at  its  extremity,  by  means  of  which  it 
may  be  ascertained  when  the  bob  and  point  upon  the  table  are 
in  the  same  vertical  line. 

302.  The  Tripod  and  its  Head  are  similar  to  those  of  the 
ordinary  transit,  though  heavier. 

A  metallic  plate,  screwed  fast  to  the  table  and  having  a  solid 
conical  spindle  projecting  from  its  centre,  affords  the  means  of 
attaching  the  head  to  the  table. 


PLANE-TABLE, 
As  MADE  BY  HELLER  &  BRIGHTLY,  PHILADELPHIA,  PA, 


THE   PLANE-TABLE.  265 

The  tripod-head  admits  of  a  slight  lateral  motion  to  the 
board,  and  is  provided  with  levelling-clamp  and  tangent- 
screws  similar  to  the  common  transit. 

303.  Of    Alidades   there   are   several   kinds.      One   of   the 
best,  however,  for  ordinarv  purposes  is  indicated  in  the  figure. 
It    consists    of    a    brass    ruler    or    straight    edge    about   22 
inches  long  and  two  inches  wide,   from  which  rises  a  column 
surmounted  by  a  telescope.     The  power   of  the   telescope  at 
least  equals  that  of  the  common  transit,  and  it  is  provided  with 
stadia   wires,    has   an   attached    level,    vertical    arc,    with   the 
necessary  adjusting  movements.     It   is    set  on  the  column  so 
that  the  line  of  collimation  is  in  or  near  the  same  vertical  plane 
with  the  bevelled  edge  of  the  ruler. 

A  parallel  ruler  allowing  a  very  slight  deviation  from  this 
plane  is  sometimes  used,  and  the  work  is  thereby  facilitated. 
A  small  level  is  placed  on  the  top  of  the  column,  which  serves 
to  indicate  any  unequal  settling  of  the  instrument.  Two  spirit 
levels  at  right  angles  to  each  other  are  placed  upon  the  table  to 
indicate  when  by  the  levelling-screws  it  is  made  horizontal ;  or, 
the  levels  are  attached  to  the  ruler  of  the  alidade,  one  in  the 
longitudinal  direction  of  the  ruler,  the  other  perpendicular 
to  it. 

304.  The  Declinator  is  simply  a  box  containing  a  magnetic 
needle  which  has  a  range  of    12  or  15  degrees  on  each  side 
of  the  zero.     It  is  used  in  orienting  the  table  ;  that  is,  to  place 
a  given   point   on    the    table  over  that  on  the  ground   which 
it  represents,  and  to  cause  a  line  of  the  paper  to  lie  in  the  same 
vertical  plane,  or  parallel  thereto,  with  its  counterpart  on  the 
ground. 

Before  the  table  is  removed  from  its  first  position,  or  at  the 
time  of  drawing  the  first  line  of  the  survey,  the  declinator  may 
be  placed  upon  it,  and  the  needle  allowed  to  rest  at  zero ;  then 
a  pencil  drawn  alongside  the  box  will  trace  a  north  and  south 
line,  since  the  sides  of  the  box  are  made  parallel  to  the  line  of 


2(36  PLANE   SURVEYING. 

zeros.*  When  the  table  is  oriented  at  any  other  station,  the 
declinator  will  give  the  same  reading  if  placed  along  the  same 
line. 

ADJUSTMENTS. 

305.  From  the  nature  of  the  service  in  some  sections  of  the 
country,  the  plane-table  is  often  necessarily  subjected  to  rough 
usage,  and  there  is  a  constant  liability  to  a  disturbance  of  the 
adjustments ;   still,  in   careful  hands,  a  well-made  instrument 
may  be  used  under  very  unfavorable  conditions  for  a  long  time 
without  being  perceptibly  affected.     One  should  not  fail,  how- 
ever,  to   make   occasional   examinations,  and   while   at  work, 
if    any  difficulty  be   encountered   which   cannot   otherwise   be 
accounted   for,  it   should   lead   directly   to  a  scrutiny  of   the 
adjustments. 

306.  The  Fiducial  Edge  of  the  Ruler.     This  should  be  a 
true,  straight  edge.     Place  the  ruler  upon  a  smooth  surface, 
and  draw  a  line  along  the  edge,  marking  also  the  lines  at  the 
ends  of  the  ruler.     Reverse  the  ruler,  and  place  the  opposite 
ends  upon  the  marked  points,  and  again  draw  the  line.      If  the 
two  lines  coincide,  no  adjustment  is  necessary  ;  if  not,  the  edge 
must  be  made  true. 

There  is  one  deviation  from  a  straight  line  which,  by  a  very 
rare  possibility,  the  edge  of  the  ruler  might  assume,  and  yet 
not  be  shown  by  the  above  test ;  it  is  when  a  part  is  convex 
and  a  part  similarly  situated  at  the  other  end  concave  in  ex- 
actly the  same  degree  and  proportion.  In  this  case,  on  reversal, 
a  line  drawn  along  the  edge  of  the  ruler  would  be  coincident 
with  the  other,  though  not  a  true  right  line  ;  this  can  be  tested 
by  an  exact  straight  edge. 

307.  The  Level  Attached  to  the  Ruler.     Place  the  instru- 
ment in  the  middle  of  the  table,  and  bring  the  bubble  to  the 
centre  by  means  of  the  levelling-screws  of  the  table  ;  draw  lines 

*  Any  other  bearing  which  may  be  read  will  answer  the  purpose. 


THE  PLANE-TABLE. 


267 


along  the  edge  and  ends  of  the  ruler  upon  the  board  to  show  its 
exact  position,  then  reverse  180°.  If  the  bubble  remain  cen- 
tral, it  is  in  adjustment;  if  not,  correct  it  one-half  by  means 
of  the  levelling-screws  of  the  table,  and  the  othei  half  by  the 
adjusting-screws  attached  to  the  level.  This  should  be  re- 
peated until  the  bubble  keeps  its  central  position,  whichever 
way  the  ruler  may  be  placed  upon  the  table.  This  presupposes 
the  plane  of  the  board  to  be  true.  If  two  levels  are  on  the 
rulers,  they  are  examined  and  adjusted  in  a  like  manner. 

Great  care  should  be  exercised  in  manipulation,  lest  the 
table  be  disturbed. 

308.  Cause  the  line  of  sight  to  revolve  in  a  vertical  plane, 
make  the  bubble  of  the  level  attached  to  the  telescope  read 
zero  when  the  line  of  sight  is  horizontal,  and  test  the  vernier 
arc  for  index  error,  each  as  in  the  transit. 


METHODS  EMPLOYED  IN  PLANE-TABLE  SURVEYING. 

309.  Points  may  be  located  with  respect  to  one  another  by 
either  of  four  methods.     In  actual  practice,  however,  a  combi- 
nation of  some  of  them  is  frequently  employed. 

310.  By  Radiation.     Suppose  it  is  required  to  make  a  plot 
of  a  field  KLMNO,  all  the  corners  of  which  can  be  SCCMI  from 
a   point  P  within  it.     Place 

the   instrument    at  P,    level 

and  clamp  it.     Find  a  point 

p    on     the     paper,    directly 

over  P  on  the  ground,  and, 

keeping    the    bevelled    edge 

of  the  ruler  on  p,  point  the 

telescope   to  any    corner    of 

the  tract,  as  K.     By  means 

of  the  stadia  wires,  or  chain,  obtain  the  distance  PK,  and  lay 

it  off  to  any  desired  scale  in  the  direction  of  the  point  sighted, 


268  PLANE   SUEVEYING. 

thus  plotting  pk.  In  a  similar  manner,  locate  the  other  corners. 
Join  by  straight  lines  the  points  thus  determined  ;  and  the 
resulting  figure  klmno  will  represent  the  tract  surveyed.  It  is 
obvious  that  the  position  of  objects  such  as  buildings,  trees, 
etc.,  if  visible,  may  be  determined  by  this  method,  and  that  it 
is  immaterial  whether  the  instrument  be  set  up  in  the  field  or 
at  one  of  the  angles,  providing  all  the  stations  can  be  seen  from 
the  point  selected. 

311.  By  Progression.  This  method  requires  the  instrument 
to  be  set  up  at  every  station  of  the  tract  to  be  surveyed.  Let 

KLMNO  represent,  as  be- 
fore, the  field,  and  suppose 
the  instrument  is  first  placed 
at  7T,  and  that  k  on  the 
paper  designates  this  point. 
With  the  alidade  directed 
towards  />,  draw  along  it  an 
indefinite  line.  Obtain  by 
stadia  or  chain  the  distance 
KL,  and  lay  it  off  to  a  desired  scale,  thus  locating  L  Remove 
the  instrument  to  L,  orient  it,  and  locate  m.  Continue  in  the 
same  manner  to  locate  n  and  o. 

When  the  table  is  oriented  at  any  station,  as  M,  the  line 
ML  should  lie  in  the  vertical  plane,  with  its  representative  ml 
on  the  plot,  and,  having  gone  round  the  tract,  the  last  line 
should  close  with  the  first  station  k. 

This  method,  in  conjunction  with  the  preceding,  may  be 
employed  advantageously  in  the  survey  of  a  road,  stream,  etc. 
The  centre  line  of  the  road  or  bank  of  the  stream  may  be  trav- 
ersed by  the  instrument,  placing  it  at  each  angle  or  bend,  as 
in  the  survey  of  a  field  by  progression,  and  determine  by  the 
method  of  radiation  the  position  of  prominent  objects,  such  as 
buildings,  bridges,  trees,  -etc.  If  there  be  added  to  the  above 
a  sketch  of  the  general  features  of  the  ground,  a  complete  map 
will  be  had  of  the  belt  of  country  traversed. 


METHODS    EMPLOYED. 


269 


312.  By  Intersections.    Let  it  be  required  to  plot  the  stations 
M,  0,  P.     Measure  carefully  the  base  line  LN,  and  draw  to  a 
convenient   scale   In   on    the    paper 

to  represent  it.  At  the  extremities 
of  this  base  line  orient  and  point  the 
instrument  to  the  several  stations. 
The  intersections  of  the  pairs  of 
lines  drawn  from  the  base  line  to 
these  stations  will  indicate  their 
position  on  the  plot.  Their  dis- 
tances from  the  base  line,  if  desired, 
may  be  obtained  by  applying  the  scale  used  in  the  construction 
of  In. 

If  a  field  or  closed  tract  of  land  is  to  be  surveyed,  a  portion 
or  all  of  one  side  may  be  used  as  a  base  line,  or  a  base  may  be 
chosen  outside  the  tract. 

This  method  is  obviously  well  adapted  to  the  mapping  of 
harbors,  shore  lines,  and  generally  to  inaccessible  points. 

Of  course  in  this,  as  in  all  triangulations,  well-conditioned 
triangles  give  more  satisfactory  results  ;  that  is  to  say,  avoid, 
if  possible,  angles  less  than  30°  or  greater  than  150°. 

313.  By  Resection.     This  method  requires  the  measurement 
of  one  line  and  the  accessibility  of  all  the  stations. 

Let  KLMNO  represent  the 
points  to  be  plotted. 

Obtain  the  distance  between 
two  of  them,  as  OK,  lay  it  off 
on  the  table  to  a  suitable 
scale,  and  let  ok  represent  it. 
Orient  the  table  at  k,  point 
the  alidade  to  L,  and  draw 
along  its  fiducial  edge  an  in- 
definite line.  Remove  the  in- 
strument to  L,  and  orient  it.  Then  with  the  alidade  centring 
on  o,  point  it  in  the  direction  of  0,  and  draw  a  line  along  its 


270  PLANE   SURVEYING. 

edge :  this  line  will  intersect  kL  in  some  point  Z,  which  will 
locate  L  on  the  plot.  Through  I  draw  a  line  towards  J/, 
remove  the  instrument  to  Jf,  and  proceed  as  before.  Objects 
on  either  side  of  the  lines  may  be  determined  by  radiation  or 
by  intersection,  and  further  details,  if  desired,  sketched  in  as 
the  work  proceeds. 

314.  Determination  of  Position  by  Resection  on  Three 
Known  Points.  In  this  problem  three  stations,  L,  N,  0,  are 
plotted,  as  Z,  ?i,  o,  on  the  table,  and  the  instrument  being  set 
up  over  a  fourth  point  P,  it  is  required  to  find  the  position  of 
this  point  on  the  map.  This  is  the  three-point  problem  of 
which  geometrical  constructions  and  analytical  solutions  are 
given  in  Chapter  II.  Section  IV.  It  may  be  solved  thus : 
Fasten  a  sheet  of  tracing-paper  on  the  board,  fix  a  point  p  to 
represent  the  station  at  which  the  instrument  is  set ;  with  the 
alidade  centring  on  p,  direct  the  telescope  successively  to  Z>,  0, 
and  jV,  and  draw  lines  of  indefinite  length  along  the  ruler's 
edge  towards  these  stations.  Then  if  the  tracing-paper  be 
shifted  until  the  three  lines  thus  drawn  coincide  with  the  points 
Z,  o,  and  »,  the  point  p  will  indicate  the  position  of  P. 

The  position  of  this  point  may  now  be  transferred,  by  pricking, 
to  the  map,  the  tracing-paper  removed,  and  the  table  oriented. 

315.*  Bessel's  Method  by  Inscribed  Quadrilateral.  A  quadri- 
lateral is  constructed  with  all  the  angles  in  the  circumference  of 
a  circle,  one  diagonal  of  wlr.ch  passes  through  the  middle  one 
of  the  three  fixed  points  and  the  point  sought.  On  this  line 
the  alidade  is  set,  the  telescope  directed  to  the  middle  point, 
and  the  table  is  in  position.  Resection  upon  the  extreme  points 
intersects  in  this  line  and  determines  the  position  of  the  point 
sought. 

Let  a,  6,  c,  be  the  points  on  the  sheet  representing  the  signals 
A,  B,  (7,  in  the  ground. 

The  table  is  set  up  at  the  point  to  be  determined  (cZ)  and 

*  Articles  315  and  316  are  from  the  U.  S.  C.  &  G.  S.  Report  for  1880. 


METHODS    EMPLOYED. 


271 


levelled.  The  alidade  is  set  upon  the  line  ca,  and  a  directed, 
by  revolving  the  table,  to  its  corresponding  signal  A,  and  the 
table  clamped ;  then,  with  the  alidade  centring  on  c,  the  mid- 
dle signal  B  is  sighted  with  the 
telescope,  and  the  line  ce  drawn 
along  the  edge  of  the  ruler. 
The  alidade  is  then  set  upon  the 
line  oc,  and  the  telescope  di- 
rected to  the  signal  (7,  by  re- 
volving the  table,  and  the  table 
clamped.  Then,  with  the  alidade 
centring  on  a,  the  .telescope  is 
directed  to  the  middle  signal  B, 
and  the  line  ae  is  drawn  along 
the  edge  of  the  ruler.  The 
point  e  (the  intersection  of  these 
two  lines)  will  be  in  the  line 
passing  through  the  middle  point  and  the  point  sought.  Set 
the  alidade  upon  the  line  be,  direct  6  to  the  signal  B  by  revolv- 
ing the  table,  and  the  table  will  be  in  position.  Clamp  the 
table,  centre  the  alidade  upon  a,  direct  the  telescope  to  the 
signal  A,  and  draw  along  the  ruler  the  line  ad.  This  will  inter- 
sect the  line  be  at  the  point  sought.  Resection  upon  (7,  cen- 
tring the  alidade  on  c  in  the  same  manner  as  upon  A,  will  verify 
its  position. 

The  opposite  angles  of  the  quadrilateral  adce  being  supple- 
mentary, angle  ace  and  angle  ade  are  subtended  by  the  same 
chord  ae  and  cae  and  cde  are  subtended  by  the  same  chord 
ce ;  consequently,  the  intersection  of  ae  and  ce  at  e  must  fall 
on  the  line  db ;  or,  the  segments  of  two  intersecting  chords 
in  a  circle  being  reciprocally  proportional,  the  triangles  ad/ and 
cef  are  similar,  and  the  triangles  cdf  and  aef  are  similar,  and 
d,  /,  and  e  must  be  in  a  right  line  passing  through  6. 

316.  Determination  of  Position  by  Resection  on  Two  Known 
Points.  This  is  called  the  two-point  problem,  there  being  given 


272  PLANE   SURVEYING. 

bv  their  projections  a,  b,  two  points  A  and  B,  to  put  the  plane- 
table  in  position  at  a  third  point  C.  (The  capital  letters  refer 
to  points  on  the  ground,  and  the  small  ones  to  their  correspond 
ing  projections.) 

Select  a  fourth  point  Z),  such  that  the  intersections  from  C 
and  D  upon  A  and  B  make  sufficiently  large  angles  for  good 
determinations.  Put  the  table  approximately  in  position  at  D, 
by  estimation  or  by  compass,  and  draw  the  lines  Aa,  Bb,  inter- 
secting in  d ;  through  d  draw  a  line  to  C.  Then  set  up  at  C, 
and  assuming  the  point  c  on  the  line  dC  at  an  estimated  dis- 
tance from  d,  and  putting  the  table  in  a  position  parallel  to  that 
which  is  occupied  at  D,  by  means  of  the  line  cd,  draw  the  lines 
from  c  to  A,  and  from  c  to  B.  These  will  intersect  the  lines  cL4, 
dB,  at  points  a'  and  b',  which  form  with  c  and  d  a  quadrilateral 
similar  to  the  true  one,  but  erroneous  in  size  and  position. 


The  angles  which  the  lines  ab  and  a'b'  make  with  each  other 
is  the  error  in  position.  By  constructing  now  through  c  a  line 
cd',  making  the  same  angle  with  cd  as  that  which  ab  makes  with 
a'b',  and  directing  this  line  cd1  to  D,  the  table  will  be  brought 
into  position,  and  the  true  point  c  can  be  found  by  the  inter- 
sections of  aA  and  bB. 

Instead  of  transferring  the  angle  of  error  by  construction,  we 
may  conveniently  proceed  as  follows,  observing  that  the  angle 
which  the  line  a'b'  makes  with  ab  is  the  error  in  the  position  of 
the  table.  As  the  table  now  stands,  a'b'  is  parallel  with  AB, 
but  we  want  to  turn  it  so  that  06  shall  be  parallel  to  the  same. 
If,  therefore,  we  place  the  alidade  on  a'b',  and  set  up  a  mark 


METHODS   EMPLOYED.  273 

in  that  direction,  then  place  the  alidade  on  ab,  and  turn  the 
table  until  it  again  points  to  the  mark,  then  ab  will  be  parallel 
to  AB,  and  the  table  is  in  position. 

317.  Practical  Suggestions  in  using  the  Plane-Table.*  The 
board  should  be  placed  so  low  as  to  be  readily  reached,  even 
at  the  most  remote  corner,  and  yet  high  enough  to  enable  the 
observer  to  take  sight  with  com  fort.  This  will  bring  it  a  little 
below  the  elbow. 

Care  must  be  taken  that  no  part  of  the  body  touch  or  rest 
against  the  edge  of  the  board.  In  using  the  alidade,  steady 
the  standard  with  the  left  hand,  while  the  right  swings  the  rear 
end  of  the  ruler  in  the  proper  direction. 

Thumb-tacks  and  rollers  for  holding  down  the  sheet  are  both 
found  objectionable,  especially  in  high  winds.  The  edges  may 
be  pasted  underneath,  or  spring  clamps  may  be  used  to  advan- 
tage. A  scale  graduated  upon  the  fiducial  edge  of  the  alidade 
is  inconvenient,  and  in  some  positious  impracticable  and  waste- 
ful of  time.  A  detached  triangular  boxwood  or  metal  scale  is 
greatly  to  be  preferred.  Umbrellas  or  shades,  whilst  a  great 
relief  to  the  eyes,  are  cumbersome  and  troublesome,  and  by 
blowing  over  on  the  table  may  cause  damage  or  derangement. 
Colored  glasses  screening  the  eyes  will  be  better,  and  by  using 
tinted  paper,  as  manilla,  instead  of  white,  still  more  relief  is 
given,  and  the  sheet  can  be  kept  cleaner. 

Before  leaving  the  station,  and  at  any  intervals  not  otherwise 
employed,  the  "  check"  shots  should  be  tesieu  to  determine  any 
displacement  of  the  board. 

Use  as  hard  a  pencil,  and  make  as  few  lines,  as  possible. 
In  locating  points  of  contours,  plot  the  distance  at  once  along 
the  edge  of  ruler  by  detached  scale,  making  only  a  dot  at  the 
point  which  should  receive  the  number  of  the  contour. 

Objects  on  a  straight  line  may  be  quickly  located  by  plotting 
the  ends  and  determining  the  intermediate  points  by  intersecting 
shots. 

*  From  The  Topographer,  by  L.  M.  Haupt,  C.E.,  Philadelphia. 


274  PLANE   SURVEYING. 

EXERCISES  WITH  THE  PLANE-TABLE. 

1.  Make  a  plane-table  survey  of  a  field,  using  one  side  as  a 
base  line. 

2.  Make  a  survey  embracing  200  or  300  rods  of  a  road  or 
stream,  locating  prominent  objects  on  either  side. 

3.  Locate  several  points  on  the  table   by  intersections,  and 
check  the  work  by  resection  from  these  points. 

4.  Locate  a  non-plotted  point  by  resection  on  three  known 
points  —  tirst  method;  check  by  Bessel's  method. 


CHAPTER  VI. 


THE  SUBVEY  OP  THE  PUBLIC  LANDS   OF  THE  UNITED 
STATES. 


THE  SOLAR  COMPASS. 

318.  A  description  of  the  Solar  Compass,  the  instrument 
that  is  extensively  used  in  the  survey  of  the  public  lands,  its 
adjustment  and  use,  will  be  given  before  describing  the  method 
employed  by  the  government  in  these  surveys. 

This  instrument,  so  ingeniously  contrived  for  readily  deter- 
mining a  true  meridian  or  north  and  south  line,  was  invented 
by  William  A.  Burt,  of  Michigan,  and  patented  by  him  in  1836. 

It  has  since  come  into  general  use  in  the  surveys  of  United 
States  public  lands,  the  principal  lines  of  which  are  required  to 
be  run  with  reference  to  the  true  meridian. 

The  arrangement  of  its  sockets  and  plates  is  similar  to  that 
of  the  Surveyor's  Transit,  as  shown  in  Chapter  II.  Section  I., 
except  that  the  sight-vanes  are  attached  to  the  under  plate  or 
limb,  and  this  revolves  around  the  upper  or  vernier  plate  on 
which  the  solar  apparatus  is  placed. 

The  limb  is  divided  to  half-degrees,  is  figured  in  two  rows, 
as  usual,  and  reads  by  the  two  opposite  verniers  to  single 
minutes. 

THE  SOLAR  APPARATUS. 

319.  The  Solar  Apparatus  is  seen  in  the  place  of  the  needle, 
and  in  fact  operates  as  its  substitute  in  the  field. 

It  consists  mainly  of  three  arcs  of  circles,  by  which  can  be 
set  off  the  latitude  of  a  place,  the  declination  of  the  sun,  and 
the  hour  of  the  day. 


276  PLANE  SURVEYING. 

These  arcs,  designated  in  the  cut  by  the  letters  a,  6,  and 
c,  are  therefore  termed  the  latitude,  the  declination,  and  the 
hour  arcs  respectively. 

320.  The  Latitude  Arc  a  has  its  centre  of  motion  in  two 
pivots,  one  of  which  is  seen  at  d  ;  the  other  is  concealed  in  the 
cut. 

It  is  moved  either  up  or  down  within  a  hollow  arc,  seen  in 
the  cut,  by  a  tangent-screw  at  /,  and  is  securely  fastened  in 
any  position  by  a  clamp-screw. 

The  latitude  arc  is  graduated  to  quarter-degrees,  and  reads 
by  its  vernier  e  to  single  minutes  ;  it  has  a  range  of  about 
35  degrees,  so  as  to  be  adjustable  to  the  latitude  of  any  place 
in  the  United  States. 

321.  The   Declination  Arc  b  is  also  graduated  to  quarter- 
degrees,  and  has  a  range  of  about  28  degrees. 

Its  vernier  w,  reading  to  single  minutes,  is  fixed  to  a  movable 
arm  ^,  having  its  centre  of  motion  at  the  end  of  the  declination 
arc  at  g ;  the  arm  is  moved  over  the  surface  of  the  declination 
arc,  and  its  vernier  set  to  any  reading  by  turning  the  head  of 
the  tangent-screw  k.  It  is  also  securely  clamped  in  any  posi- 
tion by  a  screw,  concealed  in  the  engraving. 

322.  Solar  Lenses  and  Lines.     At  each  end  of  the  arm  h  is 
a  rectangular  block  of  brass,  in  which  is  set  a  small  convex 
lens,  having  its  focus  on  the  surface  of  a  little  silver  plate  A 
(marginal  figure) ,  fastened  by  screws  to  the  inside  of  the  oppo- 
site block. 

On  the  surface  of  the  plate  are  marked  two  sets  of  lines 
intersecting  each  other  at  right  angles ;  of 
these  bb  are  termed  the  hour  lines,  and  cc  the 
^  equatorial  lines,  as  having  reference  respec- 
tively to  the  hour  of  the  day  and  the  position  of 
the  sun  in  relation  to  the  equator.  In  the  cut  the  equatorial 
lines  are  those  on  the  lower  block,  parallel  to  the  surface  of  the 


THE    SOLAB   COMPASS.  279 

hour  arc  c ;  the  hour  lines  are  of  course  those  at  right  angles 
to  the  first. 

323* Equatorial  Sights.  On  the  top  of  each  of  the  rec- 
tangular blocks  is  seen  a  little  sighting-piece,  termed  the  equa- 
torial sight,  fastened  to  the  block  by  a  small,  milled  head-screw, 
so  as  to  be  detached  at  pleasure. 

They  are  used,  as  will  be  explained  hereafter,  in  adjusting 
the  different  parts  of  the  solar  apparatus. 

324.  The  Hour  Arc  c  is  supported  by  the  two  pivots  of  the 
latitude  arc  already  spoken  of,  and  is  also  connected  with  that 
arc  by  a  curved  arm,  as  shown  in  the  figure. 

The  hour  arc  has  a  range  of  about  120°,  is  divided  to  half- 
degrees,  and  figured  in  two  series,  designating  both  the  hours 
and  the  degrees,  the  middle  division  being  marked  12  and  90 
on  either  side  of  the  graduated  lines. 

325.  The  Polar  Axis.     Through  the  centre  of  the  hour  arc 
passes  a  hollow  socket  p  containing  the  spindle  of  the  declina- 
tion arc,  by  means  of  which  this  arc  can  be  moved  from  side  to 
side  over  the  surface  of  the  hour  arc,  or  turned  completely  round, 
as  may  be  required. 

The  hour  arc  is  read  by  the  lower  edge  of  the  graduated  side 
of  the  declination  arc. 

The  axis  of  the  declination  arc,  or  indeed  the  whole  socket 
p,  is  appropriately  termed  the  polar  axis. 

326.  The  Adjuster.     Besides  the   parts  shown  in  the  cut, 
there  is  also  an  arm  used  in  the  adjustment  of  the  instrument 
as  described  hereafter,  but  laid  aside  in  the  box  when  that  is 
effected. 

The  parts  just  described  constitute  properly  the  solar 
apparatus. 

Besides  these,  however,  are  seen  the  needle-box  n  with  its 
arc  and  tangent  screw  £,  and  the  spirit  levels,  for  bringing  the 
whole  instrument  to  a  horizontal  position. 


280  PLANE   SURVEYING. 

327.  The  Needle-Box  n  has  an  arc  of  about  86  degrees  in 
extent,  divided  to  half-degrees,  and  figured  from  the  centre  or 
zero  mark  on  either  side. 

The  needle,  which  is  made  as  in  other  instruments,  except 
that  the  arms  are  of  unequal  lengths,  is  raised  or  lowered  by  a 
lever  shown  in  the  cut. 

The  needle-box  is  attached  by  a  projecting  arm  to  a  tangent- 
screw  t,  by  which  it  is  moved  about  its  centre,  and  its  needle 
set  to  any  variation. 

This  variation  is  also  read  off  by  the  vernier  on  the  end  of 
the  projecting  arm,  reading  to  three  minutes  a  graduated  arc, 
attached  to  the  plate  of  the  compass. 

328.  The  Levals  seen  with  the  solar  apparatus  have  ground- 
glass  vials,  and  are  adjustable  at  their  ends  like  those  of  other 
instruments. 

The  edge  of  the  circular  plate  on  which  the  solar  work  is 
placed  is  divided  and  figured  at  intervals  of  10  degrees,  and 
numbered,  as  shown,  from  0  to  90  on  each  side  of  the  line  of 
sight. 

These  graduations  are  used  in  connection  with  a  little  brass 
pin,  seen  in  the  centre  of  the  plate,  to  obtain  approximate 
bearings  of  lines,  which  are  not  important  enough  to  require  a 
close  observation. 

329.  Lines  of  Refraction.    The  inside  faces  of  the  sights  are 
also  graduated  and  figured,  to  indicate  the  amount  of  refraction 
to  be  allowed  when  the  sun  is  near  the  horizon. 


PRINCIPLES  OF  THE  SOLAR  COMPASS. 

330.  The  interval  between  two  equatorial  lines  cc,  in  figure 
on  page  276,  as  well  as  between  the  hour  lines  66,  is  just  suffi- 
cient to  include  the  circular  image  of  the  sun,  as  formed  by  the 
solar  lens  on  the  opposite  end  of  the  revolving  arm  ft,  figure 
on  page  277. 


THE   SOLAR   COMPASS.  281 

When,  therefore,  the  instrument  is  made  perfectly  horizontal, 
the  equatorial  lines  and  the  opposite  lenses  being  accurately 
adjusted  to  each  other  by  a  previous  operation,  and  the  sun's 
image  brought  within  the  equatorial  lines,  his  position  in  the 
heavens,  with  reference  to  the  horizon,  will  be  defined  with 
precision. 

Suppose  the  observation  to  be  made  at  the  time  of  one  of 
the  equinoxes  ;  the  arm  /t,  set  at  zero  on  the  declination  arc 
b ;  and  the  polar  axis  p,  placed  exactly  parallel  to  the  axis  of 
the  earth. 

Then  the  motion  of  the  arm  h,  if  revolved  on  the  spindle  of 
the  declination  arc  around  the  hour  circle  c,  will  exactly  corre- 
spond with  the  motion  of  the  sun  in  the  heavens,  on  the  given 
day  and  at  the  place  of  observation ;  so  that  if  the  sun's  image 
was  brought  between  the  lines  cc  in  the  morning,  it  would 
continue  in  the  same  position,  passing  neither  above  nor  below 
the  lines,  as  the  arm  was  made  to  revolve  in  imitation  of  the 
motion  of  the  sun  about  the  earth. 

In  the  morning,  as  the  sun  rises  from  the  horizon,  the  arm  h 
will  be  in  a  position  nearly  at  right  angles  to  that  shown  in  the 
cut,  the  lens  being  turned  towards  the  sun,  and  the  silver  plate 
on  which  his  image  is  thrown  directly  opposite. 

As  the  sun  ascends,  the  arm  must  be  moved  around,  until 
when  he  has  reached  the  meridian,  the  graduated  side  of  the 
declination  arc  will  indicate  12  on  the  hour  circle,  and  the  arm 
h,  the  declination  arc  b,  and  the  latitude  arc  a  will  be  in  the 
same  plane. 

As  the  sun  declines  from  the  meridian,  the  arm  h  must  be 
moved  in  the  same  direction,  until  at  sunset  its  position  will  be 
the  exact  reverse  of  that  it  occupied  in  the  morning. 

331.  Allowance  for  Declination.  Let  us  now  suppose  the 
observation  made  when  the  sun  has  passed  the  equinoctial  point, 
and  when  his  position  is  affected  by  declination. 

Bv  referring  to  the  almanac,  and  setting  off  on  the  arc  his 
declination  for  the  given  day  and  hour,  we  are  still  able  to 


282  PLANE   SURVEYING. 

determine  his  position  with  the  same  certainty  as  if  he  remained 
on  the  equator. 

When  the  sun's  declination  is  south,  that  is,  from  the  22d 
of  September  to  the  20th  of  March  in  each  year,  the  arc  b  is 
turned  towards  the  plates  of  the  compass,  as  shown  in  the  en- 
graving, and  the  solar  lens  o,  with  the  silver  plate  opposite, 
are  made  use  of  in  the  surveys. 

The  remainder  of  the  year  the  arc  is  turned  from  the  plates, 
and  the  other  lens  and  plate  employed. 

When  the  solar  compass  is  accurately  adjusted,  and  its  plates 
made  perfectly  horizontal,  the  latitude  of  the  place,  and  the 
declination  of  the  sun  for  the  given  day  and  hour,  being  also 
set  off  on  the  respective  arcs,  the  image  of  the  sun  cannot  be 
brought  between  the  equatorial  lines  until  the  polar  axis  is  placed 
in  the  plane  of  the  meridian  of  the  place,  or  in  a  position  parallel 
to  the  axis  of  the  earth.  The  slightest  deviation  from  this  posi- 
tion will  cause  the  image  to  pass  above  or  below  the  lines,  and 
thus  discover  the  error. 

We  thus,  from  the  position  of  the  sun  in  the  solar  system, 
obtain  a  certain  direction  absolutely  unchangeable,  from  which 
to  run  our  lines  and  measure  the  horizontal  angles  required. 

This  simple  principle  is  not  only  the  basis  of  the  construction 
of  the  solar  compass,  but  the  sole  cause  of  its  superiority  to  the 
ordinary  or  magnetic  instrument.  For  in  a  needle  instrument 
the  accuracy  of  the  horizontal  angles  indicated,  and  therefore 
of  all  the  observations  made,  depends  upon  "•  the  delicacy  of 
the  needle,  and  the  constancy  with  which  it  assumes  a  certain 
direction,  termed  the  magnetic  meridian." 

The  principal  causes  of  error  in  the  needle,  briefly  stated,  are 
the  dulling  of  the  pivot,  the  loss  of  polarity  in  the  needle,  the 
influence  of  local  attraction,  and  the  effect  of  the  sun's  rays, 
producing  the  diurnal  variation. 

From  all  these  imperfections  the  solar  instrument  is  free. 

The  sights  and  the  graduated  limb  being  adjusted  to  the  solar 
apparatus,  and  the  latitude  of  the  place  and  the  declination  of  the 
sun  also  set  off  upon  the  respective  arcs,  we  are  able  not  only 


THE   SOLAR   COMPASS.  283 

to  run  the  true  meridian,  or  a  due  east  and  west  course,  but 
also  to  set  off  the  horizontal  angles  with  minuteness  and  ac- 
curacy from  a  direction  which  never  changes,  and  is  unaffected 
by  attraction  of  anv  kind. 

To  ADJUST  THE  SOLAR  COMPASS. 

The  adjustments  of  this  instrument,  with  which  the  surveyor 
will  have  to  do,  are  simple  and  few  in  number,  and  will  now 
be  given  in  order. 

332.  To  Adjust  the  Levels.   Proceed  precisely  as  directed  in 
the  account  of  the  other  instruments  we  have  described',  by 
bringing  the  bubbles  into  the  centre  of  the  tubes  by  the  level- 
ling-screws  of    the  tripod,   and  then  reversing  the  instrument 
upon  its  spindle,  and  raising  or  lowering  the  ends  of  the  tubes, 
until  the  bubbles  will  remain  in  the  centre  during  a  complete 
revolution  of  the  instrument. 

333.  To  Adjust  the   Equatorial   Lines   and   Solar   Lenses. 
First  detach  the  arm  h  from  the  declination  arc  by  withdrawing 
the  screws  shown  in  the  cut  from  the  ends  of  the  posts  of  the 
tangent-screw  k,  and   also  the  clamp-screw,  and   the   conical 
pivot  with  its  small  screws  by  which  the  arm  and  declination 
arc  are  connected. 

The  arm  h  being  thus  removed,  attach  the  adjuster  in  its 
place  by  replacing  the  conical  pivot  and  screws,  and  insert  the 
clamp-screw  so  as  to  clamp  the  adjuster  at  any  point  on  the 
declination  arc. 

Now  level  the  instrument,  place  the  arm  h  on  the  adjuster, 
with  the  same  side  resting  against  the  surface  of  the  declination 
arc  as  before  it  was  detached.  Turn  the  instrument  on  its 
spindle  so  as  to  bring  the  solar  lens  to  be  adjusted  in  the  direc- 
tion of  the  sun,  and  raise  or  lower  the  adjuster  on  the  declina- 
tion arc,  until  it  can  be  clamped  in  such  a  position  as  to  bring 
the  sun's  image  as  near  as  may  be  between  the  equatorial  lines 
on  the  opposite  silver  plate,  and  bring  the  image  precisely  into 


284  PLANE   SURVEYING. 

position  by  the  tangent  of  the  latitude  arc  or  the  levelling- 
screws  of  the  tripod.  Then  carefully  turn  the  arm  half-way 
over,  until  it  rests  upon  the  adjuster  by  the  opposite  faces  of 
the  rectangular  blocks,  and  again  observe  the  position  of  the 
sun's  image. 

If  it  remains  between  the  lines  as  before,  the  lens  and  plate 
are  in  adjustment ;  if  not,  loosen  the  three  screws  which  con- 
fine the  plate  to  the  block,  and  move  the  plate  under  their 
heads,  until  one-half  the  error  in  the  position  of  the  sun's  image 
is  removed. 

Again  bring  the  image  between  the  lines,  and  repeat  the 
operation  until  it  will  remain  in  the  same  situation,  in  both 
positions  of  the  arm,  when  the  adjustment  will  be  completed. 

To  adjust  the  other  lens  and  plate,  reverse  the  arm  eud  for 
end  on  the  adjuster,  and  proceed  precisely  as  in  the  former  case, 
until  the  same  result  is  attained. 

In  tightening  the  screws  over  the  silver  plate,  care  must  be 
taken  not  to  move  the  plate. 

This  adjustment  now  being  complete,  the  adjuster  should  be 
removed,  and  the  arm  h  with  its  attachments  replaced  as 
before. 

334.  To  Adjust  the  Vernier  of  the  Declination  Arc.  Hav- 
ing levelled  the  instrument,  and  turned  its  lens  in  the  direction 
of  the  sun,  clamp  to  the  spindle,  and  set  the  vernier  v  of  the 
declination  arc  at  zero,  by  means  of  the  tangent-screw  at  &, 
and  clamp  to  the  arc. 

See  that  the  spindle  moves  easily  and  yet  truly  in  the  socket, 
or  polar  axis,  and  raise  or  lower  the  latitude  arc  by  turning  the 
tangent-screw/,  until  the  sun's  image  is  brought  between  the 
equatorial  lines  on  one  of  the  plates.  Clamp  the  latitude  arc 
by  the  screw,  and  bring  the  image  precisely  into  position  by 
the  levelling-screws  of  the  tripod  or  socket,  and  without  dis- 
turbing the  instrument,  carefully  revolve  the  arm  /«,  until  the 
opposite  lens  and  plate  are  brought  in  the  direction  of  the  sun, 
and  note  if  the  sun's  image  comes  between  the  lines  as  before. 


THE   SOLAR   COMPASS.  285 

If  it  does,  there  is  no  index  error  of  the  declination  arc ;  if 
not,  with  the  tangent-screw  &,  move  the  arm  until  the  sun's 
image  passes  over  half  the  error ;  again  bring  the  image  be- 
tween the  lines,  and  repeat  the  operation  as  before,  until  the 
image  will  occupy  the  same  position  on  both  the  plates. 

We  shall  now  find,  however,  that  the  zero  marks  on  the  arc 
and  the  vernier  do  not  correspond,  and  to  remedy  this  error, 
the  little  flat-head  screws  above  the  vernier  must  be  loosened 
until  it  can  be  moved  so  as  to  make  the  zeros  coincide,  when 
the  operation  will  be  completed. 

335.  To  Adjust  the  Solar  Apparatus  to  the  Compass  Sights. 
First  level  the  instrument,  and  with  the  clamp  and  tangent  screws 
set  the  main  plate  at  90°  by  the  verniers  and  horizontal  limb. 
Then  remove  the  clamp-screw,  and  raise  the  latitude  arc  until 
the  polar  axis  is  by  estimation  very  nearly  horizontal,  and  if 
necessary,  tighten  the  screws  on  the  pivots  of  the  arc,  so  as  to 
retain  it  in  this  position. 

Fix  the  vernier  of  the  declination  arc  at  zero,  and  direct  the 
equatorial  sights  to  some  distant  and  well-marked  object,  and 
observe  the  same  through  the  compass  sights.  If  the  same 
object  is  seen  through  both,  and  the  verniers  read  to  90°  on  the 
limb,  the  adjustment  is  complete ;  if  not,  the  correction  must 
be  made  by  moving  the  sights  or  changing  the  position  of  the 
verniers. 

To  USE  THE  SOLAR  COMPASS. 

336.  Before  this  instrument  can  be  used  at  any  given  place, 
it  is  necessary  to  set  off  upon  its  arcs  both  the  declination  of  the 
sun  as  affected  by  its  refraction  for  the  given  day  and  hour,  and 
the  latitude  of  the  place  where  the  observation  is  made. 

337.  To  Set  off  the  Declination.   The  declination  of  the  sun, 
given  in  the  ephemeris  of  the  Nautical  Almanac  from  year  to 
year,  is  calculated  for  apparent  noon  at  Greenwich,  England, 
or  Washington,  D.C. 

To  determine  it  for  any  other  hour  at  a  place  in  the  United 


286  PLANE   SUKVEYING. 

States,  reference  must  be  had,  not  only  to  the  difference  of  time 
arising  from  the  longitude,  but  also  to  the  change  of  declination 
from  day  to  day. 

By  the  use  of  standard  time,  which  is  now  quite  general 
throughout  the  United  States,  it  is  very  easy  to  obtain  the 
declination  required  at  any  place. 

For  those  using  75th  meridian  time,  a  difference  of  five  hours 
must  be  allowed  for  the  difference  in  declination  between  the 
place  of  observation  and  Greenwich. 

The  time-piece  referred  to  the  75th  meridian  as  standard  in- 
dicating 7  A.M.  when  it  is  noon  at  Greenwich. 

Where  the  90th  meridian  is  used  as  standard,  six  hours  must 
be  allowed,  etc. 

To  obtain  the  declination  for  the  other  hours  of  the  day,  take 
from  the  almanac  the  declination  for  apparent  noon  of  the  given 
clay,  and,  as  the  declination  is  increasing  or  decreasing,  add  to 
or  subtract  from  the  declination  of  the  first  hour  the  difference 
for  one  hour  as  given  in  the  ephemeris,  which  will  give,  when 
affected  by  the  refraction,  the  declination  for  the  succeeding 
hour  ;  and  proceed  thus  in  making  a  table  of  the  declination  for 
every  hour  of -the  day. 

338.  Refraction.    By  reason  of  the  increasing  density  of  the 
atmosphere  from  its  upper  regions  to  the  earth's  surface,  the 
rays  of  light  from  the  sun  are  bent  out  of  their  course,  so  as  to 
make  his  altitude  appear  greater  than  is  actually  the  case. 

The  amount  of  refraction  varies  according  to  the  altitude  of 
the  body  observed ;  being  0  when  it  is  in  the  zenith,  about  one 
minute  when  midway  from  the  horizon  to  the  zenith,  and  almost 
34'  when  in  the  horizon. 

339.  Effect  of  Incidental  Refraction.    It  will  be  seen  by 
referring   to  the  instrument,   that  the  effect  of   the   ordinary 
refraction  upon  the  position  of  the  sun's  image  with  reference 
to  the  equatorial  lines,  which,  in  fact,  are  the  only  ones  to  be 
regarded  in  running  lines  with  the  solar  compass,  is  continually 


THE  SOLAR   COMPASS.  287 

changing,  not  only  with  the  change  of  latitude,  but  also  with 
that  of  the  sun's  declination  from  hour  to  hour,  and  the  motion 
of  the  revolving  arm  as  it  follows  the  sun  in  its  daily  revolution. 

If  the  equatorial  lines  were  always  in  the  same  vertical  plane 
with  the  sun,  as  would  be  the  case  at  the  equator  at  the  time  of 
the  equinoxes,  it  is  evident  that  refraction  would  have  no  effect 
upon  the  position  of  the  image  between  these  lines,  and  there- 
fore would  not  be  of  any  importance  to  the  surveyor. 

But  as  we  proceed  further  north,  and  as  the  sun's  declination 
to  the  south  increases,  the  refraction  also  increases,  and  must 
now  be  taken  into  account. 

Again,  the  angle  which  the  equatorial  lines  make  with  the 
horizon  is  continually  changing  as  the  arm  is  made  to  follow 
the  motion  of  the  sun  during  the  course  of  a  day. 

Thus,  in  the  morning  and  evening  they  are  more  or  less 
inclined  to  the  horizon,  while  at  noon  they  are  exactly  parallel 
to  it. 

And  thus  it  follows  that  the  excess  of  refraction  at  morning 
and  evening  is  in  some  measure  balanced  by  the  fact  that  the 
position  of  the  sun's  image  with  reference  to  the  equatorial  lines 
is  then  less  affected  by  it,  on  account  of  the  greater  inclination 
of  the  lines  to  the  horizon. 

340.  Allowance  for  Refraction.  The  proper  allowance  to  be 
made  for  refraction  in  setting  off  the  declination  of  the  sun 
upon  the  solar  compass  for  any  hour  of  any  day  of  the  year  is 
given  in  the  following  table : 


288 


PLANE  SURVEYING. 


A  TABLE  OF  MEAN  REFRACTIONS  IN  DECLINATION. 

To  apply  on  the  declination  arc  of  Solar  Attachment  of  either  Compass 
or  Transits.* 


1 

DECLINATIONS. 

i 

FOR  LATITUDE  30°. 

1 

+  20° 

+  15* 

+  10D 

+  5° 

0° 

-5° 

-10^ 

-153 

-20^ 

Oh. 

10" 

15" 

21" 

27" 

33" 

40" 

48" 

57" 

1'08" 

2 

14 

19 

25 

31 

38 

46 

54 

1'05 

1  18 

3 

20 

26 

32 

39 

47 

55 

1'06 

119 

136 

4 

32 

39 

46 

52 

1'06 

1'19 

135 

157 

229 

5 

I'OO 

I'lO 

1'24 

1'52 

207 

244 

346 

543 

1306 

FOR  LATITUDE  S25  30'. 

Oh. 

13" 

18" 

24" 

30" 

36" 

44" 

52" 

1'02" 

1'14" 

2 

17 

22 

28 

35 

42 

50 

I'OO 

1  11 

126 

3 

23 

29 

35 

43 

51 

I'Ol 

1  13 

128 

147 

4 

35 

43 

51 

I'Ol 

1'IS 

127 

146 

2  13 

254 

5 

roa 

116 

1'Sl 

153 

220 

305 

425 

736 

FOB  LATITUDE  35°. 

Oh. 

15" 

21" 

27" 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

2 

20 

25 

32 

38 

46 

55 

1'05 

1  18 

135 

3 

26 

33 

39 

47 

56 

1'07 

121 

138 

200 

4 

39 

47 

56 

1'07 

1'20 

136 

1  59 

232 

325 

5 

1'07 

1*20 

1'38 

200 

234 

329 

514. 

1016 

FOR  LATITUDE  37°  30'. 

Oh. 

18" 

24" 

30" 

36" 

44" 

52" 

1'02" 

1'14'' 

1'29" 

2 

22 

28 

35 

42 

50 

I'OO 

112 

126 

145 

3 

29 

36 

43 

52 

1'02 

1  14 

129 

149 

216 

4 

43 

51 

I'Ol 

1'13 

127 

149 

2  14 

254 

405 

5 

I'll 

1'26 

145 

2  10 

249 

355 

615 

1458 

*  Computed  by  Edward   W.   Arms,   C.E.,  for  W.   and  L.   E.  Gurley 
Troy,  N.Y. 


THE   SOLAR   COMPASS. 


289 


HOUR  ANGLE. 

DECLINATIONS. 

FOB  LATITUDE  40°. 

+  203 

+  15° 

+  10° 

+  5° 

0° 

-5° 

-10° 

-15° 

-20' 

Oh. 

21" 

27" 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

1'39" 

2 

25 

32 

39 

46 

52 

1'06 

1  19 

135 

1  57 

8 

33 

40 

48 

57 

1'08 

121 

138 

202 

236 

4 

47 

55 

1'06 

1'19 

136 

158 

230 

321 

459 

5 

1'15 

1'31 

1  51 

220 

305 

425 

734 

2518 

FOR  LATITUDE  42°  30'. 

Oh. 

24" 

30" 

36" 

44" 

52" 

1'02" 

1'14" 

1'29" 

1'49" 

2 

28 

35 

39 

50 

I'OO 

1  12 

126 

145 

2  11 

3 

36 

43 

52 

1'02 

1  13 

129 

149 

2  17 

2  59 

4 

50 

I'OO 

I'll 

1  26 

1  44 

2  10 

249 

355 

6  16 

5 

1'16 

136 

158 

230 

322 

500 

924 

FOB  LATITUDE  45°. 

Oh. 

27" 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

1'39» 

2'02" 

2 

32 

39 

46 

52 

1'06 

1  19 

1  35 

1  57 

229 

3 

40 

47 

56 

1'07 

121 

138 

200 

234 

329 

4 

54 

1'04 

1'16 

133 

1  54 

224 

311 

438 

8  15 

5 

1'23 

141 

205 

241 

340 

540 

1202 

FOR  LATITUDE  4T°  30'. 

Oh. 

30" 

36" 

44" 

52" 

1'02" 

1'14" 

1'29" 

1'49" 

2'18" 

2 

35 

42 

50 

I'OO 

1  12 

1  26 

145 

201 

251 

3 

43 

61 

I'Ol 

I  13 

1  28 

147 

2  16 

2  56 

408 

4 

56 

1'09 

123 

140 

205 

240 

339 

637 

1118 

5 

1'27 

146 

212 

252 

401 

630 

1619 

FOR  LATITUDE  50°. 

•  Oh. 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

1'39" 

2'02" 

2'36" 

2 

38 

46 

55 

1'06 

1  18 

135 

157 

228 

319 

3 

47 

56 

i'oe 

1  19 

1  36 

229 

231 

323 

502 

4 

1'02 

1'14 

1  29 

1  48 

2  16 

258 

418 

669 

1947 

5 

1  30 

1  51 

2  19 

304 

422 

728 

2410 

290  PLANE   SURVEYING. 


EXPLANATION  OF  THE  TABLE  OF  REFRACTIONS.* 

The  table  is  calculated  for  latitudes  between  30°  and  50°  at 
intervals  of  2^°,  that  being  as  near  as  is  required. 

The  declination  ranges  from  0  to  20°,  both  north  and  south, 
the  +  declinations  being  north,  and  —  south,  and  is  given  for 
every  5  degrees,  that  being  sufficiently  near  for  all  practical  pur- 
poses. 

The  hour  angle  in  the  first  column  indicates  the  distance  of 
the  sun  from  the  meridian  in  hours,  the  refraction  given  for  0 
hours  being  that  which  affects  the  observed  declination  of  the 
sun  when  on  the  meridian,  commonly  known  as  meridional  re- 
fraction ;  the  refraction  for  the  hours  just  before  and  after  noon 
is  so  nearly  that  of  the  meridian,  that  it  may  be  called  and 
allowed  as  the  same. 

When  the  table  is  used,  it  must  be  borne  in  mind  that  when 
the  declination  is  north  or  +  in  the  table,  the  refraction  is  to 
be  added ;  when  the  declination  is  south  or  —  the  refraction 
must  be  subtracted. 

It  will  be  noticed  that  the  refraction  in  south  or  —  declina- 
tion increases  very  rapidly  as  the  sun  nears  the  horizon,  show- 
ing that  observations  should  not  be  taken  with  the  sun  when 
south  of  the  equator,  less  than  one  hour  from  the  horizon. 

Thus,  suppose  it  be  required  to  obtain  the  declination  for  any 
hour  in  the  day,  April  16,  1887,  at  Pittsburg,  Pa.,  where  75th 
meridian  time  is  used. 

The  difference  in  time  is  5  hours,  so  that  the  declination 
given  in  the  ephemeris  for  apparent  noon  of  that  day  at  Green- 
wich would  be  that  of  7  A.M.  at  Pittsburg.  Proceed  as  follows  : 

Declination  at  Greenwich,  mean  noon,  April  16,  1887, 
N.  10°  6'  29" 

Add  1'  51"=refract'n  for  5  hrs.  [lat.  Pittsburg  40°  28']. 

Or,  N.  10°  8'  20"  =dec.  7  A.M.  at  Pittsburg. 
*  See  also  Refraction  Table,  page  92. 


THE  SOLAR  COMPASS.  291 

To  get  the  declination  for  8  o'clock,  same  day  and  place,  add 
53",  the  difference  for  one  hour  —  because  the  declination  is 
increasing  —  to  the  declination  taken  from  the  almanac,  and 
this  increased  by  the  refraction  corresponding  to  4  hours  from 
noon  will  give  10°  8'  28"  for  the  required  declination. 

Again,  suppose  it  be  desired  to  obtain  the  corrected  dec- 
lination for  8  A.M.  Oct.  15,  1887,  same  place. 

The  declination  being  now  south,  the  refraction  is  to  be  sub- 
tracted, but  the  hourly  difference  is  to  be  added  because  the 
declination  is  increasing,  as  in  the  first  example  ;  thus  : 

Declination  at  Greenwich,  mean  noon,  Oct.  15,  1887, 

S.  8°  30'  20" 
Add  56"=  dec.  for  1  hr.,  and  increasing. 

S.  8°  31'  16" 
Subtract  2'  23"=  refr.  4  hrs.  from  noon. 


Or,  S.  8°  28'  53"=  dec.  at  8  A.M.  ; 

and  so  on  for  any  hour  in  the  day,  obtaining  from  the  declina- 
tion at  Greenwich,  by  the  proper  application  of  the  hourly 
motion,  the  declination  corresponding  to  the  hour  required,  and 
correcting  this  for  refraction  due  to  altitude. 

To  facilitate  operations,  the  calculation  of  the  declination  for 
the  different  hours  of  the  day  should  be  made  and  noted  before 
the  surveyor  commences  his  work. 

341.  To  Set  off  the  Latitude.  Find  the  declination  of  the 
sun  for  the  given  day  at  noon,  at  the  place  of  observation  as 
just  described,  and  with  the  tangent-screw  set  it  off  upon  the 
declination  arc,  and  clamp  the  arm  firmly  to  the  arc. 

Observe  in  the  almanac  the  equation  of  time  for  the  given 
day,  in  order  to  know  about  the  time  the  sun  will  reach  the 
meridian. 

Then,  about  fifteen  or  twenty  minutes  before  this  time,  set 
up  the  instrument,  level  it  carefully,  fix  the  divided  surface  of 
the  declination  arc  at  12  on  the  hour  circle,  and  turn  the  instru- 


292  PLA^E   SURVEYING. 

ment  upon  its  spindle  until  the  solar  lens  is  brought  into  the 
direction  of  the  sun. 

Loosen  the  clamp-screw  of  the  latitude  arc,  and  with  the 
tangent-screw  raise  or  lower  this  arc  until  the  image  of  the  sun 
is  brought  precisely  between  the  equatorial  lines,  and  turn  the 
instrument  from  time  to  time  so  as  to  keep  the  image  also 
between  the  hour  lines  on  the  plate. 

As  the  sun  ascends,  its  image  will  move  below  the  lines,  and 
the  arc  must  be  moved  to  follow  it.  Continue  thus,  keeping  it 
between  the  two  sets  of  lines  until  its  image  begins  to  pass 
above  the  equatorial  lines,  which  is  also  the  moment  of  its  pass- 
ing the  meridian. 

Now  read  off  the  vernier  of  the  arc,  and  we  have  the  latitude 
of  the  place,  which  is  always  to  be  set  off  on  the  arc  when  the 
compass  is  used  at  the  given  place. 

It  is  the  practice  of  surveyors  using  the  solar  compass  to  set 
off,  in  the  manner  just  described,  the  latitude  of  the  point  where 
the  survey  begins,  and  to  repeat  the  observation  and  correction 
of  the  latitude  arc  every  day  when  the  weather  is  favorable, 
there  being  also  an  hour  at  mid-day  when  the  sun  is  so  near  the 
meridian  as  not  to  give  the  direction  of  lines  with  the  certainty 
required. 

342.  To  Run  Lines  with  the  Solar  Compass.  Having  set 
off  in  the  manner  just  given  the  latitude  and  declination  upon 
their  respective  arcs,  the  instrument  being  also  in  adjustment, 
the  surveyor  is  ready  to  run  lines  by  the  sun. 

To  do  this,  the  instrument  is  set  over  the  station  and  care- 
fully levelled,  the  plates  clamped  at  zero  on  the  horizontal  limb, 
and  the  sights  directed  north  and  south,  the  direction  being 
given,  when  unknown,  approximately  by  the  needle. 

The  solar  lens  is  then  turned  to  the  sun,  and  with  one  hand 
on  the  instrument,  and  the  other  on  the  revolving  arm,  both 
are  moved  from  side  to  side,  until  the  sun's  image  is  made  to 
appear  on  the  silver  plate  ;  when,  by  carefully  continuing  the 
operation,  it  may  be  brought  precisely  between  the  equatorial 
lines. 


THE   SOLAR   COMPASS.  293 

Allowance  being  now  made  for  refraction,  .the  line  of  sights 
will  indicate  the  true  meridian  ;  the  observation  may  now  be 
made,  and  the  flag-man  put  in  position. 

When  a  due  east  and  west  line  is  to  be  run,  the  verniers  of 
the  horizontal  limb  are  set  at  90°,  and' the  sun's  image  kept 
between  the  lines  as  before. 

The  solar  compass  being  so  constructed  that  when  the  sun's 
image  is  in  position  the  limb  must  be  clamped  at  0  in  order  to 
run  a  true  meridian  line,  it  will  be  evident  that  the  bearing  of 
any  line  from  the  meridian  may  be  read  by  the  verniers  of  the 
limb  precisely  as  in  the  ordinary  magnetic  compass :  the  bear- 
ings of  lines  are  read  from  the  ends  of  the  needle. 

343.  Use  of  the  Needle.     In  running  lines,  the  magnetic 
needle  is  always  kept  with  the  sun  ;   that  is,  the  point  of  the 
needle  is  made  to  indicate  0  on  the  arc  of  the  compass-box  by 
turning  the  tangent-screw  connected  with  its  arm  on  the  oppo- 
site side  of  the  plate.     By  this  means  the  lines  can  be  run  by 
the  needle  alone  in  case  of  the  temporary  disappearance  of  the 
sun  ;  but,  of  course,  in  such  cases  the  surveyor  must  be  sure 
that  no  local  attraction  is  exerted. 

The  variation  of  the  needle,  which  is  noted  at  every  station, 
is  read  off  in  degrees  and  minutes  on  the  arc,  by  the  edge  of 
which  the  vernier  of  the  needle-box  moves. 

344.  Allowance  for  the  Earth's  Curvature.    When  long  lines 
are  run  by  the  solar  compass,  either  by  the  true  meridian,  or 
due  east  and  west,  allowance  must  be  made  for  the  curvature 
of  the  earth. 

Thus,  in  running  north  or  south,  the  latitude  changes  about 
one  minute  for  every  distance  of  92  chains  30  links,  and  the 
side  of  a  township  requires  a  change  on  the  latitude  arc  of  5' 
12",  the  township,  of  course,  being  six  miles  square. 

This  allowance  is  of  constant  use  where  the  surveyor  fails 
to  get  an  observation  on  the  sun  at  noon,  and  is  a  very  close 
approximation  to  the  truth. 


294  PLANE   SURVEYING. 

In  running  due  east  and  west,  as  in  tracing  the  standard 
parallels  of  latitude,  the  sights  are  set  at  90°  on  the  limb,  and 
the  line  is  run  at  right  angles  to  the  meridian. 

If  no  allowance  were  made  for  the  earth's  curvature,  these 
lines  would,  if  sufficiently  produced,  reach  the  equator,  to 
which  they  are  constantly  tending. 

Of  course,  in  running  short  lines  either  east  or  west,  the 
variation  from  the  parallel  would  be  so  small  as  to  be  of  no 
practical  importance ;  but  when  long  sights  are  taken,  the 
correction  should  be  made  by  taking  fore  and  back  sights  at 
every  station,  noticing  the  error  on  the  back-sight,  and  setting 
off  one-half  of  it  on  the  fore-sight  on  the  side  towards  the  pole. 

345.  Time  of  Day  by  the  Sun.      The  time  of  day  is  best 
ascertained    by    the   solar   compass   when    the   sun    is   on   the 
meridian,  as  at  the  time  of  making  the  observation  for  lati- 
tude. 

The  time  thus  given  is  that  of  apparent  noon,  and  can  be 
reduced  to  mean  time,  by  merel}'  applying  the  equation  of  time 
as  directed  in  the  almanac,  and  adding  or  subtracting  as  the 
sun  is  slow  or  fast. 

The  time,  of  course,  can  also  be  taken  before  or  after  noon, 
by  bringing  the  sun's  image  between  the  hour  lines,  and 
noticing  the  position  of  the  divided  edge  of  the  revolving 
arm,  with  reference  to  the  graduations  of  the  hour  circle, 
allowing  four  minutes  of  time  for  each  degree  of  the  arc,  and 
thus -obtaining  apparent  time,  which  must  be  corrected  by  the 
equation  of  time  as  just  described. 

346.  Caution  as  to  the  False  Image.     In  using  the  compass 
upon  the  sun,  if  the  revolving  arm  be  turned  a  little  one  side  of 
its  proper  position,  a  false  or  reflected  image  of  the  sun  will 
appear  on  the  silver  plate  in  nearly  the  same  place  as  that  occu- 
pied by  the  true  one.     It  is  caused  by  the  reflection  of  the  true 
image  from  the  surface  of  the  arm,  and  is  a  fruitful  source  of 
error   to  the  inexperienced   surveyor.      It   can,    however,    be 


SURVEY   OF  THE   PUBLIC   LANDS.  295 

readily  distinguished  from  the  real  image  by  being  much  less 
bright,  and  not  so  clearly  defined. 

347.  Approximate  Bearings.     When  the  bearings  of  lines, 
such  as  the  course  of  a  stream,  or  the  boundaries  of  a  forest, 
are  not  desired  with  the  certainty  given  by  the  verniers  and 
horizontal  limb,  a  rough  approximation  of  the  angle  they  make 
with  the    true   meridian  is  obtained   by  the  divisions  on  the 
outside  of  the  circular  plate. 

In  this  operation,  a  pencil,  or  thin  straight  edge  of  any  sort, 
is  held  perpendicularly  against  the  circular  edge  of  the  plate, 
and  moved  around  until  it  is  in  range  with  the  eve,  the  brass 
centre-pin,  and  the  object  observed. 

The  bearing  of  the  line  is  then  read  off  at  the  point  where  the 
pencil  is  placed. 

348.  Time  for  Using  the  Solar  Compass.     The  solar  com- 
pass, like  the  ordinary  instrument,  can  be  used  at  all  seasons 
of  the  year,  the  most  favorable  time  being,  of  course,  in  the 
summer,  when  the  declination  is  north,  and  the  days  are  long, 
and  more  generally  fair.* 

ORIGIN  OP  THE  SYSTEM  FOR  THE  SURVEY  OF  THE 
PUBLIC   LANDS.f 

349.  The  present  system  of  survey  of  the  public  lands  was 
inaugurated    by   a   committee    appointed   by   the   Continental 
Congress,   of  which  Thomas   Jefferson  was   chairman.      This 
committee,  on  May  7,  1784,  reported  an  ordinance  requiring 
public  lands  to  be  divided  into  "  hundreds"  of  ten  geographical 
miles  square,  and  these  again  subdivided  into  lots  of  one  mile 
square,  each  to  be  numbered  from  1  to  100,  commencing  in  the 
northwestern  corner  and  continuing  from  west  to  east  and  from 


*  See  Article  147. 

t  The  following  pages  regarding  the  government  surveys  are  from 
"  Instructions  of  the  General  Land  Office  to  the  Surveyors-General  of  the 
United  States  relative  to  the  Survey  of  the  Public  Lands." 


296 


PLANE   SUKVEYING. 


east  to  west  consecutively.  By  subsequent  amendment,  April 
26,  1785,  the  ordinance  required  the  surveyors  "  to  divide  the 
said  territory  into  townships  of  7  miles  square,  by  lines  running 
due  north  and  south,  and  others  crossing  these  at  right  angles. 
The  plots  of  the  townships,  respectively,  shall  be  marked  by 
subdivisions  into  sections  of  1  mile  square,  or  640  acres  in  the 
same  direction  as  the  external  lines,  and  numbered  from  1  to 
49,  and  these  sections  shall  be  subdivided  into  lots  of  320 
acres."  This  is  the  first  record  of  the  use  of  the  terms  "  town- 
ship "  and  "  section." 

This  ordinance  was  subsequently  still  further  amended,  and 
as  finally  passed  on  the  20th  of  May,  1785,  provided  for  town- 
ships 6  miles  square,  containing  36  sections  of  1  mile  square. 
The  first  public  surveys  were  made  under  this  ordinance  by 
the  direction  of  the  Geographer  of  the  United  States. 


6 

5 

4 

5 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

82 

33 

34 

35 

36 

The  act  of  Congress,  approved  May  18,  1796,  provided 
for  the  appointment  of  a  surveyor-general,  and  directed 
the  survey  of  lands  northwest  of  the  Ohio  River,  and  above 
the  mouth  of  the  Kentucky  River,  "  in  which  the  titles  of  the 
Indian  tribes  have  been  extinguished,"  and  among  other  pro- 
visions, that  the  "  sections  shall  be  numbered  respectively, 


SURVEY   OF  THE   PUBLIC   LAMDS.  297 

beginning  with  the  number  one  in  the  northeast  section  and 
proceeding  west  and  east,  alternately,  through  the  township, 
with  progressive  numbers  till  the  thirty-sixth  be  completed." 
This  method  of  numbering  sections,  as  shown  by  the  preceding 
diagram,  is  still  in  use. 

The  act  of  Congress,  approved  Feb.  11,  1805,  directs  the 
subdivisions  of  the  public  lands  into  quarter-sections.  The 
act  of  April  24,  1820,  provides  for  the  sale  of  the  public  lands 
in  half-quarter-sections,  and  that  in  every  case  of  the  division 
of  a  quarter-section,  the  division  line  shall  run  north  and  south. 
April  5,  1832,  Congress  directed  the  subdivision  of  the  public 
lands  into  quarter-quarters,  and  requiring  the  division  line  to 
run  east  and  west. 

350.  A    surveyor-general    for    each    surveying    district    is 
appointed  by  the  President,  by  and  with  the  advice  of  the 
Senate.     He  is  required,  while  in  the  discharge  of  the  duties 
of  his  office,  to  reside  in  the  district  for  which  he  is  appointed. 
His  term  of  office  is  four  years,  and  he  must  give  bonds,  with 
sufficient  security  for  the  penal  sum  of  $30,000,  for  the  faithful 
disbursement  of  all  public  money  placed  in  his  hands,  and  for 
the  faithful  performance  of  the  duties  of  his  office.     Among 
other  duties  prescribed  by  law  and  set  forth  in  the  manual,  the 
surveyor-general  is  required  to  engage  a  sufficient  number  of 
skilful  surveyors   as  his  deputies,  and  shall  cause  to  be  sur- 
veyed, measured,   and   marked,  without   delay,  all   base   and 
meridian   lines  through  such  points,  and  perpetuated  by  such 
monuments,  and  such  other  correction  parallels  and  meridians, 
as   may  be   prescribed   by  law,  or   by  instructions   from   the 
General  Land  Office,  in  respect  to  the  public  lands  within  his 
surveying  district  to  which  the  Indian  title  has  been  or  may  be 
extinguished. 

351.  System  of  Rectangular  Surveying.     The  public  lands 
of  the  United  States  are  ordinarily  surveyed  into  rectangular 
tracts,  bounded  by  lines  conforming  to  the  cardinal  points. 


298  PLANE   SURVEYING. 

The  public  lands  shall  be  laid  off,  in  the  first  place,  into 
bodies  of  land  24  miles  square,  as  near  as  may  be.  This  shall 
be  done  by  the  extension  of  standard  lines  from  the  principal 
meridian  every  24  miles,  and  by  the  extension  from  the  base 
and  standard  lines,  of  auxiliary  meridians  every  24  miles. 
Thereafter  they  shall  be  laid  off  into  bodies  of  land  6  miles 
square,  as  near  as  may  be,  called  townships,  containing,  as 
near  as  may  be,  23,040  acres.  The  townships  shall  be  sub- 
divided into  36  tracts,  called  sections,  each  containing,  as  near 
as  may  be,  640  acres.  Any  number  or  series  of  contiguous 
townships,  situate  north  or  south  of  each  other,  constitute  a 
range. 

(a)  The  law  requires  that  the  lines  of  the  public  surveys 
shall  be  governed  by  the  true  meridian,  and  that  the  township 
shall  be  six  miles  square,  —  two  things  involving  in  connection 
a  mathematical  impossibility.  For  strictly  to  conform  to  the 
meridian  necessarily  throws  the  township  out  of  square,  by 
reason  of  the  convergency  of  meridians,  and  hence  by  adhering 
to  the  true  meridian  results  the  necessity  of  departing  from  the 
strict  requirements  of  law,  as  respects  the  precise  area  of  town- 
ships and  the  subdivisions!  parts  thereof;  the  township  assum- 
ing something  of  a  trapezoidal  form,  which  inequality  develops 
itself  more  and  more  as  such,  the  higher  the  latitude  of  the 
surveys.  It  is  doubtless  in  view  of  these  circumstances  that 
the  law  provides  (see  Section  2  of  the  act  of  May  18,  1796) 
that  the  section  of  a  mile  square  shall  contain  the  quantity  of 
640  acres,  as  nearly  as  may  be;  and,  morever,  provides  (see 
Section  3  of  the  act  of  May  10,  1800)  in  the  following  words : 
"And  in  all  cases  where  the  exterior  lines  of  the  townships 
thus  to  be  subdivided  into  sections  or  half-sections  shall  exceed, 
or  shall  not  extend,  6  miles,  the  excess  or  deficiency  shall  be 
specially  noted,  and  added  to  or  deducted  from  the  western  or 
northern  ranges  of  sections  or  half-sections  in  such  township, 
according  as  the  error  may  be  in  running  the  lines  from  east  to 
west  or  from  south  to  north  ;  the  sections  and  half-sections 
bounded  on  the  northern  and  western  lines  of  such  townships 


SURVEY    OF   THE   PUBLIC    LANl)S. 


299 


shall  be  sold  as  containing  only  the  quantity  expressed  in  the 
returns  and  plats,  respectively,  and  all  others  as  containing  the 
complete  legal  quantity." 

Sections  5  and  6  of  Township  No.  6,  North,  Range  No.  34, 
east,  of  the  principal  meridian,  Montana,  are  exhibited  below : 


(5)  The  section  lines  are  surveyed  from  south  to  north  on 
true  meridians,  and  from  east  to  west,  in  order  to  throw  the 
excesses  or  deficiencies  in  measurements  on  the  north  and  west 
sides  of  the  township,  as  required  by  law.  In  a  case  where  a 
township  has  been  partially  surveyed,  and  it  is  necessary  to 
complete  the  survey  of  the  same,  or  where  the  character  of  the 
land  is  such  that  only  the  north  or  west  portions  of  the  town- 
ship can  be  surveyed,  this  rule  cannot  be  strictly  adhered  to ; 
but  in  such  cases  must  be  departed  from  only  so  far  as  is 
absolutely  necessary.  It  will  also  be  necessary  to  depart  from 
this  rule  where  surveys  close  upon  State  or  Territorial  bound- 
aries, or  upon  survej's  extending  from  different  meridians. 

(c)  The  townships  are  to  bear  numbers  in  respect  to  the 
base  line,  either  north  or  south  of  it ;  and  the  tiers  of  townships 
called  "ranges"  will  bear  numbers  in  respect  to  the  meridian 
line,  according  to  their  relative  position  to  it,  either  on  the  east 
or  west. 


300  PLANE   SURVEYING. 

(d)  The  36   sections   into  which  a  township  io  subdivided 
are  numbered,  commencing  with  number  one  at  the  northeast 
angle  of  the  township  and  proceeding  west  to  number  6,  and 
thence   proceeding  east  to  number  12,  and  so  on,  alternately 
until  the  number  36  is  in  the  southeast  angle.      In  all  cases 
of  surveys  of  fractional  townships,   the  sections  should  bear 
the  same  numbers  as  they  would  if  the  township  were  full. 

(e)  Standard  parallels  shall  be  established  at  intervals  of 
every  24  miles,  north  and  south  of  the  base  line,  and  auxiliary 
meridians  at  intervals  of  every  24  miles,  east  and  west  of  the 
principal  meridian  ;  the  object  being  to  confine  the  errors  result- 
ing from  convergence  of  meridians  and  inaccuracies  in  measure- 
ments, within  the  tracts  of  land  bounded  by  the  lines  so  estab- 
lished. 

(/)  The  survey  of  all  principal  base  and  meridian  standard 
parallels,  and  auxiliary  meridian  and  township  lines  must  be 
made  with  an  instrument  operating  independently  of  the  mag- 
netic needle.  Burt's  improved  solar  compass,  or  other  instru- 
ment of  equal  utility,  must  be  used  of  necessity  in  such  cases ; 
and  it  is  deemed  best  that  such  instrument  should  be  used  under 
all  circumstances.  Where  the  needle  can  be  relied  on,  however, 
the  ordinary  compass  may  be  used  in  subdividing  and  meander- 
ing. Whenever  deputies  use  instruments  with  magnetic  appa- 
ratus only,  they  must  test  the  accuracy  of  their  work  and  the 
condition  of  their  instruments  by  at  least  three  observations 
upon  a  circumpolar  star,  upon  different  days,  between  the  com- 
mencement and  close  of  surveying  operations  in  any  given 
township.  Deputies  using  instruments  with  solar  apparatus  are 
not  required  to  make  observations  of  the  star  Polaris,  but  they 
must  test  their  instruments  by  taking  the  latitude  daily,  weather 
permitting,  in  running  base,  standard,  meridian,  and  range 
lines,  and  upon  three  different  days,  during  the  execution  of 
subdivisional  surveys  in  each  township.  They  must  make  com- 
plete records  in  their  field  notes,  under  proper  dates,  of  the 
making  of  all  observations  in  compliance  with  these  instructions, 
showing  the  style  and  condition  of  the  instrument  in  use,  and 


SURVEY   OF   THE   PUBLIC    LANDS.  301 

the  angle  formed  by  comparing  the  line  run  with  the  meridian 
as  determined  by  observations. 

(g)  The  construction  and  adjustments  of  all  surveying  in- 
struments used  in  the  surveying  of  the  public  lands  of  the 
United  States  must  be  tested  at  least  once  a  year,  and  oftener 
if  necessary,  by  comparison  with  the  true  meridian,  established 
under  the  direction  of  the  surveyor-general  of  the  district ;  and 
the  instruments  must  be  so  modified  in  construction,  or  in  such 
a  way  corrected,  as  may  be  necessary  to  produce  the  closest 
possible  approximation  to  accuracy  and  uniformity  in  the 
operation  of  all  such  instruments.  A  record  will  be  made  of 
such  examinations,  showing  the  number  and  style  of  the  instru- 
ment, name  of  the  maker,  the  quantity  of  instrumental  error 
discovered  by  comparison,  in  either  solar  or  magnetic  apparatus, 
or  both,  and  means  taken  for  correction.  The  surveyor-general 
will  allow  no  surveys  to  be  made  until  the  instruments  to  be 
used  therefor  have  been  approved  by  him. 

(ft)  The  township  lines  and  the  subdivision  lines  will  usually 
be  measured  by  a  two-pole  chain  of  33.03  feet  in  length,  con- 
sisting of  50  links,  and  each  link  being  7.92  inches  long.  On 
uniform  and  level  ground,  however,  the  four-pole  chain  may  be 
used.  The  measurements  will,  however,  always  be  represented 
according  to  the  four-pole  chain  of  100  links.  The  four-pole 
chains  must  be  adjusted  to  lengths  of  66.06  feet,  The  object 
in  adding  six-hundredtlis  of  a  foot  to  the  66  feet  of  a  four-pole 
chain  is  to  assure  thereby  that  66  feet  will  be  set  off  upon  the 
earth's  surface  without  the  application  of  a  greater  strain  than 
about  20  pounds  by  the  chainmen,  thus  providing  for  loss  by 
vertical  curvature  of 'the  chain,  and  at  the  same  time  avoiding 
the  uncertain  results  attending  the  application  of  strains  taxing 
its  elasticity.  The  deputy  surveyor  must  provide  himself  with 
a  measure  of  the  standard  chain  kept  at  the  office  of  the  sur- 
veyor-general, to  be  used  by  him  as  a  field  standard.  The  chain 
in  use  must  be  compared  and  adjusted  with  this  field  standard 
each  working  day ;  and  such  field  standard  must  be  returned  to 
the  surveyor-general's  office  for  examination  when  his  work  is 
'•ompleted. 


302  PLANE   SURVEYING. 

352.  Of  Tally-Pins.    You  will  use  11  tally-pins  made  of  steel, 
not  exceeding  14  inches  in  length,  weighty  enough  toward  the 
point  to  make  them  drop  perpendicularly,  and  having  a  ring  at 
the  top,  in  which  is  to  be  fixed  a  piece  of  red  cloth,  or  some- 
thing else  of  conspicuous  color,  to  make  them  readily  seen  when 
stuck  in  the  ground. 

353.  Process  of  Chaining.     In  measuring  lines  with  a  two- 
pole  chain,  every  five  chains  are  called  a  tally ;  and  in  measur- 
ing lines  with  a  four-pole  chain,  every  ten  chains  are  called  a 
tally,  because  at  that  distance  the  last  of  the  10  tally-pins  with 
which  the  forward  chainman  set  out  will  have  been  stuck.     He 
then  cries  "tally";  which  cry  is  repeated  by  the  other  chain- 
man, and  each  registers  the  distance  by  slipping  a  thimble,  but- 
ton, or  ring  of  leather,  or  something  of  the  kind,  on  a  belt  worn 
for  that  purpose,  or  by  some  other  convenient  method.     The 
hind  chainman  then  comes  up,  and  having  counted  in  the  pres- 
ence of  his  fellow  the  tally-pins  which  he  has  taken  up,  so  that 
both  may  be  assured  that  none  of  the  pins  have  been  lost,  he 
then  takes  the  forward  end  of  the  chain,  and  proceeds  to  set 
the  pins.     Thus  the  chainmeu  alternately  change  places,  each 
setting  the  pins  that  he  has  taken  up,  so  that  one  is  forward  in 
all  the  odd,  and  the  other  in  all  the  even,  tallies.     Such  pro- 
cedure, it  is  believed,  tends  to  insure  accuracy  in  measurement, 
facilitates  the  recollection  of   the  distances  to  objects  on  the 
line,  and  renders  a  mis-tally  almost  impossible. 

354.  Levelling  the  Chain  and  Plumbing  the  Pins.    The  length 
of  every  line  you  run  is  to  be  ascertained  by  precise  horizontal 
measurement,  as  nearly  appi'oximating  to  an  air  line  as  is  possi- 
ble in  practice  on  the  earth's  surface.     This  all-important  object 
can  only  be  attained  by  a  rigid  adherence  to  the  three  following 
observances : 

Ever  keeping  the  chain  stretched  to  its  utmost  degree  of  ten- 
sion on  even  ground. 

On  uneven  ground,  keeping  the  chain  not  only  stretched  as 


SURVEY    OF   THE   PUBLIC    LANDS.  303 

aforesaid,  but  horizontally  levelled.  And  when  ascending  or 
descending  steep  ground,  hills,  or  mountains,  the  chain  will  have 
to  be  shortened  to  one-half  its  length  (and  sometimes  more),  in 
order  accurately  to  obtain  the  true  horizontal  measurement. 

The  careful  plumbing  of  the  tally-pins,  so  as  to  attain  pre- 
cisely the  spot  where  they  should  be  stuck.  The  more  uneven 
the  surface,  the  greater  the  caution  needed  to  set  the  pins. 

355.  Marking  Lines.     All  lines  on  which  are  to  be  estab- 
lished the  legal  corner  boundaries  are  to  be  marked  after  this 
method,  viz.  :  Those  trees  which  may  intercept  the  line  must 
have  two  chops  or  notches  on  each  side  of  them,  without  any 
other  marks  whatever.     These  are  called  sight  trees  or  line  trees. 
A  sufficient  number  of  other  trees  standing  within  50  links  of 
the  line,  on  either  side  of    it,  are  to  be  blazed  on  two  sides 
diagonally,  or  quartering  toward  the  line,  in  order  to  render  the 
line   conspicuous,  and    readily  to  be  traced,  the  blazes  to  be 
opposite  each  other,  coinciding  in  direction  with  the  line  where 
the  trees  stand  very  near  it,  and  to  approach  nearer  each  other 
the  farther  the  line  passes  from  the  blazed  trees. 

Where  trees  two  inches  or  more  in  diameter  are  found,  the 
required  blazes  must  not  be  omitted. 

Bushes  on  or  near  the  line  should  be  bent  at  right  angles 
therewith,  and  receive  a  blow  of  the  axe  at  about  the  usual 
height  of  blazes  from  the  ground  sufficient  to  leave  them  in  a 
bent  position,  but  not  to  prevent  their  growth. 

356.  On  Trial  or  Random  Lines   the  trees   are   not  to  be 
blazed,  unless  occasionally,  from  indispensable  necessity,  and 
then  it  must  be  done  so  guardedly  as  to  prevent  the  possibility^ 
of  confounding  the  marks  of  the  trial  line  with  the  true.     But 
bushes  and  limbs  of  trees  may  be  lopped,  and  stakes  set  on  the 
trial  or  random  line,  at  every  ten  chains,  to  enable  the  surveyor 
on  his  return  to  follow  and  correct  the  trial  line,  and  establish 
therefrom  the  true  line.     To  prevent  confusion,  the  temporary 
stakes  set  on  the  trial  or  random  lines  must  be  pulled  up  when 
the  surveyor  returns  to  establish  the  true  line. 


304  PLANE   SURVEYING. 

357.  Insuperable  Objects  on  Line ;   Witness  Points.     Under 
circumstances  where  your  course  is  obstructed  by  impassable 
obstacles,    such    as    ponds,    swamps,    marshes,    lakes,    rivers, 
creeks,  etc.,  you  will  prolong  the  line  across  such  obstacles 
by  means  of  right-angle  offsets  ;    or,  if  such  be  inconvenient, 
by  a  traverse  or  trigonometrical  operation,  until  you  regain  the 
line  on  the  opposite  side.     And  in  case  a  north  and  south,  or  a 
true  east  and  west,  line  is  regained  in  advance  of   any  such 
obstacle,  you  will  prolong  and  mark  the  line  back  to  the  ob- 
stacle so  passed,  and  state  all  the  particulars  in  relation  thereto 
in  your  field-book.     And  at  the  intersection  of  lines  with  both 
margins  of  impassable  obstacles  you  will  establish  a  witness 
point  (for  the  purpose  of  perpetuating  the  intersections  there- 
with) ,  by  setting  a  post,  and  giving  in  your  field-book  the  course 
and  distance  therefrom  to  two  trees  on  opposite  sides  of  the 
line,  each  of  which  trees  you  will  mark  with  a  blaze  and  notch 
facing  the  post ;  but  on  the  margins  of  navigable  watercourses 
or  navigable  lakes  you  will  mark  the  trees  with  the  proper 
number  of  the  fractional  section,  township,  and  range. 

358.  The  Best  Marking-Tools  adapted  to  the  purpose  must 
be  provided  for  marking  neatly  and  distinctly  all  the  letters  and 
figures  required  to  be  made  at  corners,  Arabic  figures  being 
used  exclusively ;  and  the  deputy  is  always  to  have  at  hand  the 
necessary  implements  for  keeping  his  marking-tools  in  order. 

359.  Establishing  Corners.     To  procure  the  faithful  execu- 
tion of  this  portion  of  a  surveyor's   dutv  is  a  matter  of   the 
utmost  importance.      After  a  true   coursing   and   most   exact 
measurement,  the  establishment  of  corners  is  the  consummation 
of  the  work.     If,  therefore,  the  corners  be  not  perpetuated  in  a 
permanent  and  workmanlike  manner,  the  great  aim  of  the  sur- 
veying service  will  not  have  been  attained. 

The  following  are  the  different  points  for  perpetuating  cor- 
ners, viz.  : 

(a)  For  township  boundaries,  at  intervals  of  every  6  miles. 


SURVEY   OF   THE   PUBLIC    LANDS.  305 

(6)  For  section  boundaries,  at  intervals  of  every  mile,  or  80 
chains. 

(c)  For  quarter-section  boundaries,  at  intervals  of  every  half- 
mile,  or  40  chains.     Exceptions,  however,  occur,  as  fully  set 
forth  hereafter  in  that  portion  of  the  manual  showing  the  man- 
ner of  running  township  lines  and  method  of  subdividing. 

(d)  Meander  corners  are  established  at  all  those  points  where 
the  lines  of  the  public  surveys   intersect  the  banks  of  such 
rivers,  bayous,  lakes,  or  islands,  as  are  by  law  directed  to  be 
meandered. 

360.  Miscellaneous.  When  a  rock  in  place  is  established  for 
a  corner,  its  dimensions  above  ground  must  be  given,  and  a 
cross  (  X  )  marked  at  exact  corner  point. 

Where  mounds  of  earth  are  raised  "  alongside  "  of  corners  on 
N.  and  S.  lines,  they  must  be  placed  on  the  W.,  and  on  the  E. 
and  W.  lines  on  the  N.  side  of  corner.  In  case  the  character 
of  the  land  is  such  that  this  cannot  be  done,  the  deputy  will 
state  in  his  notes  instead  of  "  alongside"  "  S."  (on  E.). 

In  case  where  pits  are  practicable,  the  deputy  prefers  raising 
a  mound  of  stone,  or  stone  covered  with  earth,  as  more  likely 
to  perpetuate  the  corner ;  he  will  use  the  form  given  for  mound 
of  stone,  omitting  the  words  "pits  impracticable,"  and  adding 
"  covered  with  earth,"  when  so  established. 

Where  the  requisite  number  of  trees  can  be  found  within  300 
links  of  the  corner  point,  three  (3)  bearing  trees  should  be 
established  for  every  standard  or  closing  corner,  four  (4)  for 
every  corner  common  to  four  townships  or  sections,  and  two  (2) 
for  every  quarter-section  corner  or  meander  corner.  In  case  the 
requisite  number  cannot  be  found  within  limits,  the  deputy 
must  state  in  his  field  notes,  after  describing  those  established, 
"  no  other  trees  within  limits,"  and  "dug  pits  in  sees.  —  &  — ," 
or  "  raised  a  mound  of  stone  alongside." 

Stones  18  inches  and  less  long  must  be  set  two-thirds,  and 
over  18  inches  long,  three-fourths,  of  their  length  in  the  ground. 
No  stones  containing  less  than  504  cubic  inches  must  be  used 


306  PLANE   SURVEYING. 

for  corners.  Particular  attention  is  called  to  the  "  summary  of 
objects  and  data  required  to  be  noted,"  on  pages  —  and  —  of 
these  instructions,  and  it  is  expected  that  the  deputy  will  thor- 
oughly comply  with  the  same  in  his  work  and  field  notes. 

No  mountains,  swamp  lands,  or  lands  not  classed  as  survey- 
able,  are  to  be  meandered,  and  all  lines  approaching  such  lands 
must  be  discontinued  at  the  section  or  quarter-section  corner. 

Where,  by  reason  of  impassable  objects,  the  south  boundary 
of  a  township  cannot  be  established,  an  east  and  west  line 
should  be  run  through  the  township,  first  random,  and  then 
corrected,  from  one  range  line  to  the  other,  and  as  far  south  as 
possible,  and  from  such  line  the  section  lines  will  be  extended 
in  the  usual  manner,  except  over  any  fractions  south  of  said 
line,  which  may  be  surveyed  in  the  opposite  direction  from  the 
section  corners  on  the  auxiliary  base  thus  established. 

When  no  part  of  the  east  or  west  boundaries  can  be  run, 
both  north  and  south  boundaries  will  be  established  as  true 
lines.  Allowance  for  the  convergency  of  meridians  must  be 
made  whenever  necessary.* 

All  letters  and  figures  cut  in  posts  or  trees  must  be  marked 
over  with  red  chalk  to  make  them  still  more  plain  and  durable. 
Township  corners  common  to  four  townships,  and  section  cor- 
ners common  to  four  sections,  are  to  be  set  diagonally  in  the 
earth,  with  the  angles  in  the  direction  of  the  lines.  All  other 
corners  are  to  be  set  square,  with  the  sides  facing  the  direction 
of  the  lines.  The  sizes  of  wooden  posts,  mounds,  and  pits, 
noted  in  foregoing  descriptions  of  corners,  are  to  be  regarded 
as  minimum,  and  whenever  practicable  to  increase  their  dimen- 
sions, it  is  desirable  to  do  so.  In  establishing  corners,  stones 
should  be  used  whenever  practicable ;  then  posts ;  and  lastly, 
mounds,  with  stake  in  pit. 

It  is  expected  that  deputy  surveyors  will  carefully  read  and 
familiarize  themselves  with  these  instructions,  and  all  others 


*  See   Table    of   Convergency  of   Meridians  at  end   of    chapter,  and 
explanation  of  same. 


SURVEY   OF  THE   PUBLIC   LANDS.  307 

contained  in  this  volume,  and  will  instruct  their  assistants  as  to 
their  duties  before  commencing  work.  Extra  copies  will  be 
furnished  the  deputies  for  the  use  of  their  assistants. 

361.  Standard    Quarter-Section  Corners  on   standard  lines 
must  be  established  in  all  respects  like  other  quarter-section 
corners,  with  the  addition  of  the  letters  S.C.  ;   and  if  bearing 
trees  are  established  for  such  corners,  each  tree  must  be  marked 
S.C.  \  S.B.T.     When  a  pit  is  dug  at  a  meander  corner,  it  must 
be  8  links  from  the  corner  on  the  side  opposite  the  river  or  lake 
meandered. 

The  letters  M.C.,  for  "  meander  corner,"  must  be  marked  on 
the  side  facing  the  river  or  lake  meandered. 

362.  A  Witness  Corner,  in  addition  to  the  marks  that  would 
be  placed  upon  the  corner  for  which  it  is  a  witness,  must  have 
the  letters  W.C.,  and  be  established  in  all  respects  like  such 
corner. 

If  bearing  trees  are  established  for  a  witness  corner,  each 
tree  must  be  marked  W.C.,  in  addition  to  the  usual  marks. 

363.  Meandering.     Both  banks  of  navigable  rivers  are  to  be 
meandered  by  taking  the  general  courses  and  distances  of  their 
sinuosities. 

At  those  points,  when  either  the  township  or  section  lines 
intersect  the  banks  of  a  navigable  stream,  corners  are  to  be 
established  at  the  time  of  running  these  lines.  These  are 
called  meander  corners;  and  in  meandering,  you  are  to 
commence  at  one  of  these  corners,  coursing  the  banks,  and 
measuring  the  distance  of  each  course  from  your  commencing 
corner  to  the  next  meander  corner.  By  the  same  method,  you 
are  to  meander  the  opposite  bank  of  the  same  river.  The 
crossing  distance  between  meander  corners  on  same  line  is  to  be 
ascertained  by  triang illation,  that  the  river  may  be  accurately 
protracted.  Rivers  not  classed  under  the  statute  as  navigable, 
but  which  are  well-defined  natural  arteries  of  internal  communi- 
cation, will  only  be  meandered  on  one  bank. 


308  PLANE   SURVEYING. 

All  lakes,  bayous,  and  deep  ponds  which  may  serve  as  public 
highways  of  commerce  must  be  meandered. 

364.  Surveying.     Initial  points,  from  which  the  lines  of  the 
public  surveys  are  to  be  extended,  must  be  established  when- 
ever necessary  under  special  instructions,  as  may  be  prescribed 
in  each  case  by  the  Commissioner  of  the  General  Land  Office. 
The  locus  of  such  initial  points  must  be  selected  with  great 
care   and   due   consideration   for   their    prominence    and   easy 
identification,  and  must  be  established  astronomically. 

The  initial  point  having  been  established,  the  lines  of  the 
public  surveys  are  to  be  extended  therefrom  as  follows : 

365.  Base  Line.     The  base  line  shall  be  extended  east  and 
west  from  the  initial  point  by  the  use  of  solar  instruments  or 
transits,   as  may  be  directed   by  the   surveyor-general  in  his 
special  written  instructions.    Where  solar  instruments  are  used, 
the  deputy  must  test  said  instruments  in  every  12  miles  of  line 
run,  by  taking  the  latitude,  or  by  observation  on  the  polar  star ; 
and  in  all  cases  where  he  has  reason  to  suppose  that  said  instru- 
ment is  in  error,  he  must  take  an  observation  on  the  polar  star ; 
and  if  error  be  found,  must  make  the  necessary  corrections 
before  proceeding  with  his  survey.     The  proper  corners  shall 
be  established  at  each  40  and  80  chains,  and  at  the  intersection 
of  the  line  with  rivers,  lakes,  or  bayous  that  should  be  mean- 
dered, in  accordance  with  the  instructions  for  the  establishment 
of  corners.     In  order  to  check  errors  in  measurement,  two  sets 
of  chainmen,  operating  independently  of  each  other,  must  be 
employed. 

Where  transits  are  used,  the  line  will  be  run  by  setting  off  at 
the  point  of  departure  on  the  principal  meridians  a  tangent 
to  the  parallel  of  latitude,  which  will  be  a  line  falling  at  right 
angles  to  the  said  meridian.  The  survey  will  be  continued  on 
this,  line  for  twelve  (12)  miles,  but  the  corners  will  be  estab- 
lished at  the  proper  points  by  offsets  northerly  from  said  line, 
at  the  end  of  each  half-mile.  In  order  to  offset  correctly  from 


SURVEY  OF  THE   PUBLIC   LANDS.  309 

the  tangent  to  the  parallel,  the  deputy  will  be  guided  by  the  table 
of  offsets  and  azimuths  contained  in  the  Manual  of  Instructions. 
As  the  azimuth  of  the  tangent  is  shown,  the  angle  thence  to 
the  true  meridian  at  each  mile  is  readily  found,  thus  indicating 
the  direction  of  the  offset  line.  The  computations  are  made  for 
a  distance  of  12  miles,  at  the  end  of  which  observations  on  the 
polar  star  must  be  taken  for  the  projection  of  a  new  tangent. 
The  computations  are  also  upon  even  degrees  of  latitude ;  off- 
sets for  intervening  parallels  can  be  readily  determined  by 
interpolation.  Where  offset  distances  quarter-section  corners 
exceed  50  links,  their  direction  to  the  parallel  can  be  deter- 
mined in  like  manner  by  interpolation  for  azimuth.  When  said 
distances  are  less  than  50  links,  interpolation  for  determining 
the  distances  will  not  be  required. 

366.  Principal  Meridian.     The  principal  meridian  shall  be 
extended  north  and  south  from  the  initial  point,  by  the  use  of 
solar  instruments  or  transits,  as  may  be  directed  by  the  sur- 
veyor-general in  his  special  written  instructions. 

Where  solar  instruments  are  used,  the  line  will  be  run  in  the 
same  manner  as  prescribed  for  running  the  base  line  by  solar 
instruments.  Where  transits  are  used,  observations  upon  the 
polar  star  must  be  taken  within  each  12  miles  of  line  run.  In 
addition  to  the  above  general  instructions,  it  is  required  that  in 
all  cases  where  the  establishment  of  a  new  principal  meridian 
seems  to  be  necessary  to  the  surveyor-general,  he  shall  submit 
the  matter,  together  with  his  reasons  therefor,  to  the  Commis- 
sioner of  the  General  Land  Office,  and  the  survey  of  such  prin- 
cipal meridian  shall  not  be  commenced  until  written  authority, 
together  with  such  special  instructions  as  he  may  deem  neces- 
sary, shall  have  been  received  from  the  Commissioner. 

367.  Standard    Parallels.      Standard   parallels,   which    are 
also  called  correction  lines,  shall  be  extended  east  and  west 
from   the   principal   meridian,  at  intervals  of  every  24   miles 
north  and  south  of  the  base  line,  in  the  same  manner  as  pre- 
scribed for  running  the  base  line. 


310  PLANE   SURVEYING. 

Auxiliary  Meridians.  Auxiliary  meridians  shall  be  extended 
north  and  south  from  the  base  line,  at  intervals  of  every  24 
miles  east  and  west  from  the  principal  meridian,  in  the  same 
manner  as  prescribed  for  running  the  principal  meridian. 

It  is  contemplated  that  these  base,  principal  meridian,  stand- 
ard, and  auxiliary  meridian  lines  shall  first  be  extended  over 
the  territory  to  be  surveyed,  and  that  afterwards  township  and 
section  lines  shall  be  run,  where  needed,  within  these  tracts  of 
24  miles  square,  formed  by  the  extension  of  these  principal 
lines  ;  and  each  surveyor-general  will  therefore  cause  said  prin- 
cipal lines  to  be  extended  as  rapidly  as  practicable. 

368.  Exteriors,  or  Township  Lines.  The  east  and  west 
boundaries  of  townships  are  always  to  be  run  from  south  to 
north  on  a  true  meridian  line  ;  and  the  north  and  south  boun- 
daries are  to  be  run  from  east  to  west,  or  from  west  to  east 
(according  to  the  relation  of  the  township  to  be  surveyed  with 
reference  to  prior  surveys) ,  on  a  random  or  trial  line,  and  cor- 
rected back  on  a  true  line.  The  distance  north  or  south  of  the 
township  corner  to  be  closed  upon,  from  the  point  of  intersec- 
tion of  these  random  lines  with  the  east  or  west  boundary  of 
the  township,  must  be  carefully  measured  and  noted.  Should 
it  happen,  however,  that  such  random  line  should  fall  short,  or 
overrun  in  length,  or  intersect  the  east  or  west  boundary  more 
than  three  chains'  distance  from  the  township  corner  thereon,  as 
compared  with  the  corresponding  boundary  on  the  south  (due 
allowance  being  made  for  convergency)  the  line,  and  if  neces- 
sary the  entire  exterior  boundaries  of  the  township,  must  be 
retraced,  so  as  to  discover  and  correct  the  error.  In  running 
random  lines,  temporary  corners  are  to  be  set  at  each  40  and 
80  chains,  and  permanent  corners  established  upon  the  true 
line  as  corrected  back,  in  accordance  with  instructions,  throw- 
ing the  excess  or  deficiency  on  the  west  half-mile,  as  prescribed 
by  law.  Permanent  corners  are  to  be  established,  in  accord- 
ance with  instructions,  on  the  east  and  west  township  bound- 
aries at  the  time  they  are  to  be  run.  Whenever  practicable, 


SURVEY  OF  THE  PUBLIC   LANDS.  311 

the  township  lines  within  these  tracts  of  24  miles  square,  must 
be  surveyed  in  regular  order  from  south  to  north;  i.e.,  the 
exterior  boundaries  of  the  township,  in  any  one  range  lying 
immediately  north  of  the  south  boundary  of  such  tract  of  24 
miles  square,  must  first  be  surveyed,  and  the  exteriors  of  the 
other  three  townships  in  said  range  extended  therefrom,  in 
regular  order,  from  south  to  north  ;  and  it  is  preferable  to  sur- 
vey first  the  entire  range  of  townships  in  such  tract  adjoining 
the  east  boundary,  or  adjoining  the  west  boundary,  and  the 
other  three  ranges  in  regular  sequence.  In  cases,  however, 
where  the  character  of  the  land  is  such  that  this  rule  cannot  be 
complied  with,  the'  following  will  be  observed.  In  extending 
the  south  or  north  boundaries  of  a  township  to  the  west,  where 
the  southwest  or  northeast  corners  cannot  be  established  in  the 
regular  way  by  running  a  north  and  south  line,  such  boundaries 
will  be  run  west  on  a  true  line,  allowing  for  convergency  on  the 
west  half-mile ;  and  from  the  township  corner  established  at 
the  end  of  such  boundary,  the  west  boundary  will  be  ruu  north 
or  south,  as  the  case  may  be.  In  extending  south  or  north  of  a 
township  to  the  east,  where  the  southeast  or  northeast  corner 
cannot  be  established  in  the  regular  way,  the  same  rule  will  be 
observed,  except  that  such  boundaries  will  be  run  east  on  a  true 
line,  and  the  ea.si.boundary  run  north  or  south,  as  the  case  may  be. 
One  set  of  chainmen  only  is  required  in  running  township  lines. 

369.  Method  of  Subdividing.  The  first  mile,  both  on  the 
south  and  east  boundaries  of  each  township  you  are  required  to 
subdivide,  is  to  be  carefully  traced  and  measured  before  you 
enter  upon  the  subdivision  thereof.  This  will  enable  you  to 
observe  any  change  that  may  have  taken  place  in  the  magnetic 
variation  as  it  existed  at  the  time  of  running  the  township 
lines,  and  will  also  enable  you  to  compare  your  chaining  with 
that  upon  the  township  lines. 

Any  discrepancy  arising  either  from  a  change  in  the  magnetic 
variation  or  a  difference  in  measure  is  to  be  carefully  noted  in 
the  field  notes. 


312  PLANE   SURVEYING. 

After  adjusting  your  compass  to  a  variation  which  you  have 
just  found  will  retrace  the  eastern  boundary  of  the  township, 
you  will  commence  at  the  corner  to  Sections  35  and  36,  on  the 
south  boundary,  and  run  a  line  parallel  to  the  range  line,  40 
chains,  to  the  quarter-section  corner,  which  you  are  to  establish 
between  Sections  35  and  36  ;  continuing  on  said  course  40 
chains  farther,  you  will  establish  the  corner  to  Sections  25,  26, 
35,  and  36. 

From  the  section  corner  last  named,  run  a  random  line,  with- 
out blazing,  due  east,  for  the  corner  of  sections  25  and  36,  on 
east  boundary,  and  at  40  chains  from  the  starting-point  set  a 
post  for  temporary  quarter-section  corner.  If  you  intersect 
exactly  at  the  corner,  you  will  blaze  your  random  line  back, 
and  establish  it  as  the  true  line  ;  but  if  your  random  line  inter- 
sects the  said  east  boundary  either  north  or  south  of  said  corner, 
you  will  measure  the  distance  of  such  intersection,  from  which 
you  will  calculate  a  course  that  will  run  a  true  line  back  to  the 
corner  from  which  your  random  started.  You  will  establish 
the  permanent  quarter-section  corner  at  a  point  equidistant 
from  the  two  terminations  of  the  true  line. 

From  the  corner  of  Sections  25,  26,  35,  and  36,  run  due  north 
between  Sections  25  and  26,  setting  the  quarter-section  post,  as 
before,  at  40  chains,  and  at  80  chains  establishing  the  corner 
of  Sections  23,  24,  25,  and  26.  Then  run  a  random  due  east 
for  the  corner  of  Sections  24  and  25  on  east  boundary  ;  setting 
temporary  quarter-section  post  at  40  chains ;  correcting  back, 
and  establishing  permanent  quarter-section  corner  at  the  equi- 
distant point  on  the  true  line,  in  the  manner  directed  on  the 
line  between  Sections  25  and  36. 

In  this  manner  you  will  proceed  with  the  survey  of  each  suc- 
cessive section  in  the  first  tier  until  you  arrive  at  the  north 
boundary  of  the  township,  which  you  will  reach  in  running  up  a 
random  line  between  Sections  1  and  2.  If  this  random  line 
should  not  intersect  at  the  corner  established  for  Sections  1 ,  2, 
35,  and  36,  upon  the  township  line,  you  will  note  the  distance 
that  you  fall  east  or  west  of  the  same,  from  which  distance  you 


SUKVEY   OF   THE   PUBLIC    LANDS.  313 

will  calculate  a  course  that  will  run  a  true  line  south  to  the 
corner  from  which  your,  random  started.  If  the  north  boundary 
of  a  township  is  a  base  or  standard  line,  the  line  between  Sections 
1  and  2  is  to  be  run  north  as  a  true  line,  and  the  closing  corner 
established  at  the  point  of  intersection  with  such  base  or  stand- 
ard line ;  and  in  such  case,  the  distance  from  said  closing 
corner  to  the  nearest  section  or  quarter-section  corner  on  such 
base  or  standard  line  must  be  carefully  measured  and  noted  as 
a  "  connection  line." 

In  like  manner  proceed  with  the  survey  of  each  successive 
tier  of  sections  until  you  arrive  at  the  fifth  tier ;  and  from  each 
section  corner  which  you  establish  upon  this  tier  you  are  to 
run  random  lines  to  the  corresponding  corners  established  upon 
the  range  line  forming  the  western  boundary  of  the  township ; 
setting  as  you  proceed  each  temporary  quarter-section  corner 
at  40  chains  from  the  interior  section  corner,  so  as  to  throw 
the  excess  or  deficiency  of  measurement  on  the  extreme  tier  of 
quarter-sections  contiguous  to  the  township  boundary  ;  and  on 
returning  establish  the  true  line,  and  establish  thereon  the  per- 
manent quarter-section  corner. 

It  is  not  required  that  the  deputy  shall  complete  the  survey 
of  the  first  tier  of  sections  from  north  to  south  before  commenc- 
ing the  survey  of  the  second  or  any  subsequent  tier,  but  the 
corner  on  which  the  random  line  closes  must  have  been  pre- 
viously established  by  running  the  line  north  on  which  it  is 
established,  except  as  follows :  where  it  is  impracticable  to 
establish  such  section  corner  in  the  regular  manner,  it  may  be 
established  by  running  the  east  and  west  line  east  or  west,  as 
the  case  may  be,  on  a  true  line,  setting  the  quarter-section 
corner  at  40  chains  and  the  section  corner  at  80  chains. 

Quarter-section  corners,  both  upon  north  and  south  and  upon 
east  and  west  lines,  are  to  be  established  at  a  point  "  equi- 
distant" from  the  corresponding  section  corners,  except  upon 
the  lines  crossing  on  the  north  and  west  boundaries  of  the  town- 
ship, and  in  those  situations  the  quarter-section  corners  will 
always  be  established  at  precisely  40  chains  to  the  north  or 


314  PLANE   SURVEYING. 

west,  as  the  case  may  be,  of  the  respective  section  corners 
from  which  those  lines  respectively  start,  by  which  procedure 
the  excess  or  deficiency  in  the  measurements  will  be  thrown, 
according  to  law,  on  the  extreme  tier  of  quarter-sections. 

370.  Prescribed  Limits  for  Closing,  and  Length  of  Lines  in 
Certain  Cases.     Every  north-and-south  section  line,  except  those 
terminating  in  the  north  boundary  of  the  township,  must  be  80 
chains  in  length. 

The  east-and-west  section  lines,  except  those  terminating  in 
the  west  boundary  of  the  township,  are  to  be  within  80  links  of 
the  actual  distance  established  on  the  south  boundary  line  of 
the  township  for  the  width  of  said  tier  of  sections,  and  must 
close  within  80  links  north  or  south  of  the  section  corner. 

The  north  boundary  and  south  boundary  of  any  one  section, 
except  in  the  extreme  western  tier,  are  to  be  within  80  links 
of  equal  length. 

The  meanders  within  each  fractional  section,  or  between  two 
meander  posts,  or  of  an  island  in  the  interior  of  a  section,  must 
close  within  1  chain  and  50  links. 

In  running  random  township  exteriors,  if  such  random  lines 
fall  short  or  overrun  in  length  or  intersect  the  eastern  or  west- 
ern boundary,  as  the  case  may  be,  of  the  township  at  more  than 
3  chains  north  or  south  of  the  true  corner,  the  lines  must  be 
retraced,  even  if  found  necessary  to  measure  the  meridional 
boundaries  of  the  township.  One  set  of  chain  men  only  is 
required  in  subdividing. 

371.  Subdivision  of  Sections.     Under  the  provisions  of  the 
act  of  Congress  approved  Feb.  11,  1805,  the  course  to  be  pur- 
sued in  the  subdivision  of  sections  is  to  run  straight  lines  from 
the  established  quarter-section  corners  —  United  States  surveys 
—  to  the  opposite  corresponding  corners,  and  the  point  of  inter- 
section of  the  lines  so  run  will  be  the  corner  common  to  the 
several  quarter-sections  ;  or,  in  other  words,  the  legal  centre  of 
the  section. 


SURVEY   OF   THE   PUBLIC   LANDS.  315 

In  the  subdivision  of  fractional  quarter-sections  where  no 
opposite  corresponding  sections  have  been  or  can  be  fixed,  the 
subdivision  lines  should  be  ascertained  by  running  from  the 
established  corners  due  north,  south,  east,  or  west  lines,  as 
the  case  may  be,  to  the  watercourse,  Indian  boundary  line,  or 
other  external  boundary  of  such  fractional  section.  The  law 
presupposes  the  section  lines  surveyed  and  marked  in  the  field 
by  the  United  States  deputy  surveyors  to  be  due  north  and 
south  or  east  and  west  lines,  but  in  actual  experience  this  is  not 
always  the  case  ;  hence,  in  order  to  carry  out  the  spirit  of  the 
law,  it  will  be  necessary  in  running  the  subdivisioual  lines 
through  fractional  sections  to  adopt  mean  courses  where  the 
section  lines  are  not  due  lines,  or  to  run  the  subdivision  line 
parallel  to  the  section  line  where  there  is  no  opposite  section 
line. 

Upon  the  lines  closing  on  the  north  and  west  boundaries  of  a 
township  the  quarter-section  corners  are  established  by  the 
United  States  deputy  surveyors  at  precisely  40  chains  to  the 
north  or  west  of  the  last  interior  section  corners,  and  the  excess 
or  deficiency  in  the  measurement  is  thrown  on  the  outer  tier  of 
lots,  as  per  act  of  Congress  approved  May  10,  1800.  In  the 
subdivision  of  quarter-sections,  the  quarter-quarter  corners  are 
to  be  placed  at  points  equidistant  between  the  section  and 
quarter-section  corners,  and  between  the  quarter  corners  and 
the  common  centre  of  the  section,  except  on  the  last  half-mile 
of  the  lines  closing  on  the  north  or  west  boundaries  of  a 
township,  where  they  should  be  placed  at  20  chains,  propor- 
tionate measurement,  to  the  north  or  west  of  the  quarter- 
section  corner. 

The  subdivisional  lines  of  fractional  quarter-sections  should 
be  run  from  points  on  the  section  lines  intermediate  between 
the  section  and  quarter-section  corners  due  north,  south,  east, 
or  west,  to  the  lake,  watercourse,  or  reservation  which  renders 
such  tracts  fractional. 

When  there  are  double  sets  of  section  corners  on  township 
and  range  lines,  the  quarter  corners  for  the  sections  south  of  the 


816  FLANE   SURVEYING. 

township  lines  and  east  of  the  range  lines  are  not  established 
in  the  field  by  the  United  States  surveyors,  but  in  subdividing 
such  sections  said  quarter  corners  should  be  so  placed  as  to  suit 
the  calculations  of  the  areas  of  the  quarter-sections  adjoining  the 
township  boundaries  as  expressed  upon  the  official  plot,  adopt- 
ing proportionate  measurements  where  the  present  measure- 
ments of  the  north  or  west  boundaries  of  the  sections  differ 
from  the  original  measurements. 

372.  Re-establishment  of  Lost  Corners.     The  original  cor- 
ners, when  they  can  be  found,  must  stand  as  the  true  corners 
they  were  intended  to  represent,  even  though  not  exactly  where 
strict  professional  care   might   have  placed  them    in  the   first 
instance. 

As  has  been  observed,  no  existing  original  corner  can  be  dis- 
turbed, and  it  will  be  plain  that  any  excess  or  deficiency  in 
measurements  between  existing  corners  cannot  in  any  degree 
affect  the  distances  beyond  said  existing  corners,  but  must  be 
added  or  subtracted  proportionately  to  or  from  the  intervals 
embraced  between  the  corners  which  are  still  standing. 

373.  Summary  of  Objects  and  Data  required  to  be  Noted. 
The  precise  length  of  every  line  run,  noting  all  necessary  off- 
sets therefrom,  with  the  reason  and  mode  thereof. 

The  kind  and  diameter  of  all  bearing  trees,  with  the  course 
and  distance  of  the  same  from  their  respective  corners,  and  the 
precise  relative  position  of  ivitness  corners  to  the  true  corners. 

The  kind  of  materials  of  which  corners  are  constructed. 

Trees  on  line.  The  name,  diameter,  and  distance  on  line  to 
all  trees  which  it  intersects. 

Intersections  by  line  of  land  objects.  The  distance  at  which 
the  line  first  intersects  and  then  leaves  every  settler's  claim  and 
improvements;  prairie,  river,  creek,  or  other  "  bottom ";  or 
swamp,  marsh,  grove,  and  windfall,  with  the  course  of  the 
same  at  both  points  of  intersection  ;  also  the  distances  at  which 
you  begin  to  ascend,  arrive  at  the  top,  begin  to  descend,  and 


SURVEY   OF    THE   PUBLIC    LANDS.  31? 

reach  the  foot  of  all  remarkable  hills  and  ridges,  with  their 
courses,  and  estimated  height,  in  feet,  above  the  level  land  of 
the  surrounding  country,  or  above  the  bottom  lands,  ravines, 
or  waters  near  which  they  are  situated. 

Intersection  by  line  of  water  objects. 

All  rivers,  creeks,  and  smaller  streams  of  water  which  the 
line  crosses  ;  the  distances  on  line  at  the  points  of  intersection  ; 
and  their  icidtlis  on  line.  In  cases  of  navigable  streams,  their 
width  will  be  ascertained  between  the  meander  corners,  as  set 
forth  under  the  proper  head. 

The  land's  surface  —  whether  level,  rolling,  broken,  or  hilly. 

The  soil  —  whether  first,  second,  third,  or  fourth  rate. 

Timber  —  the  several  kinds  of  timber  and  undergrowth,  in 
the  order  in  which  they  predominate. 

Bottom  lands  —  to  be  described  as  wet  or  dry ;  and  if  sub- 
ject to  inundation,  state  to  what  depth. 

Springs  of  water — whether  fresh,  saline,  or  mineral,  with 
the  course  of  the  stream  flowing  from  them. 

Lakes  and  ponds  —  describing  their  banks  and  -giving  their 
height,  and  also  depth  of  water,  and  whether  it  be  pure  or 
stagnant. 

Improvements — towns  and  villages  ;  houses  or  cabins  ;  fields, 
or  other  improvements ;  sugar-tree  groves,  sugar  camps,  mill 
seats,  forges,  and  factories. 

Coal  bank  or  beds  ;  peat  or  turf  grounds  ;  minerals  and  ores, 
with  particular  description  of  the  same  as  to  quality  and  extent, 
and  all  diggings  therefor ;  also  salt  springs  and  licks.  All 
reliable  information  you  can  obtain  respecting  these  objects, 
whether  they  be  on  your  immediate  line  or  not,  is  to  appear 
on  the  general  description  to  be  given  at  the  end  of  the  notes. 

Roads  and  trails,  with  their  directions  whence  and  whither. 

Rapids,  cataracts,  cascades,  or  falls  of  water,  with  the  esti- 
mated height  of  their  fall  in  feet. 

Precipices,  caves,  sink  holes,  ravines,  stone  quarries,  ledges 
of  rocks,  with  the  kind  of  stone  they  afford. 

Natural  curiosities,  interesting  fossils,  petrifactions,  organic 


318  PLANE   SURVEYING. 

remains,  etc. ;  also  all  ancient  works  of  art,  such  as  mounds, 
fortifications,  embankments,  ditches,  or  objects  of  like  nature. 

The  variation  of  the  needle  must  be  noted  at  all  points  or 
places  on  the  lines  where  there  is  found  any  material  change 
of  variation  ;  and  the  positions  of  such  points  must  be  perfectly 
identified  in  the  notes. 

Besides  the  ordinary  notes  taken  on  line  (and  which  must 
always  be  written  down  on  the  spot,  leaving  nothing  to  be  sup- 
plied by  memory),  the  deputy  will  subjoin,  at  the  conclusion  of 
his  book,  such  further  description  or  information  touching  any 
matter  or  thing  connected  with  the  township  (or  other  survey) 
which  he  may  be  able  to  afford,  and  may  deem  useful  or  neces- 
sary to  be  known,  with  a  general  description  of  the  township  in 
the  aggregate,  as  respects  the  face  of  the  country,  its  soil  and 
geological  features,  timber,  minerals,  waters,  etc. 

374.  Specimen  Field  Notes  of  the  survey  of  the  Third 
Standard  Parallel  North,  through  Range  No.  21  east,  of  the 
principal  base  and  meridian  in  the  Territory  of  Montana,  as 
surveyed  by  James  Page,  U.  S.  Deputy  Surveyor. 

On  the  night  of  August  22,  1880,  I  took  observation  on  the 
star  Polaris,  in  accordance  with  instructions  contained  in  the 
"  Manual  of  Surveys,"  and  drove  pickets  on  the  line  thus  estab- 
lished. 

Survey  commenced  August  23,  1880,  with  a  Burt's  Improved 
Solar  Compass. 

Before  commencing  this  survey,  I  test  my  compass  on  the 
line  established  last  night,  and  find  it  correct.  I  begin  at  the 
standard  corner  to  townships  13  north,  ranges  20  and  21  east, 
which  is  a  post,  4  inches  square,  marked : 

S.C.,  T.  13  N.,  on  N. ;  R.  21  E.,  S.  31,  on  E. ;  and  R. 
20  E.,  S.  36,  on  W.  faces,  with  6  notches  on  N.,  E.,  and  W. 
faces,  and  pits  N.,  E.,  and  W.  of  post,  6  ft.  dist.,  and  mound 
of  earth  around  post. 

Thence  I  run 


SURVEY   OF   THE   PUBLIC   LANDS. 


319 


East,  on  S.  boundary  Sec.  31. 

Variation  20£°  E. 
Ascend. 

A  point  about  200  ft.  above  township  cor.  top  of  ridge. 
Set  a  sandstone  18  x  8  x  5  ins.,  12  ins.  in  the 
ground,  for  standard  ^  sec.  cor.  marked  S.C.  \  on 
N.  face ;  dug  pits  18x18x12  ins.  E.  and  W.  of 
stone,  51-  ft.  dist.,  and  raised  a  mound  of  earth 
1^  ft.  high.  31  ft.  base  alongside  ;  thence 
Enter  pine  timber. 

Set  a  sandstone  24  x  10  x  7  ins.,  18  ins.  in  the  ground 
for  standard  cor.  to  sees.  31  and  32,  marked  S.C. 
with  5  notches  on  E.  and  1  notch  on  W.  edges ; 
from  which 

A  pine,  12  ins.  diam.,  bears  N.  77°  E.,  41  Iks.  dist., 
marked  T.  13  N.,  R.  21  E.,  S.  32  B.T.  ; 

A  pine,  18  ins.  diarn.,  bears  N.  50°  W.,  20  Iks.  dist., 
marked  T.  13  N.,  R.  21  E.,  S.  31  B.T.  ; 

A  pine,  7  ins.  diam.,  bears  S.  30°  W.,  119  Iks.  dist., 
marked  T.  12  N.,  R.  21  E.,  S.  5  B.T. 

Land,  high,  mountainous,  hilly,  and  rolling. 

Soil,  sandy,  gravel,  and  rocky  ;  4th  rate. 

Timber,  pine,  23  chs.  ;  mostly  dead  and  fallen. 


East  on  S.  boundary  Sec.  32. 
Through  timber. 
Va.  20|°  E. 

Ravine,  course  S.,  about  30  ft.  deep. 

Ravine,  course  S.  20°  E.,  about  20  ft.  deep. 

Set  a  sandstone,  18  x  14  X  5  ins.,  12  ins.  in  the  ground, 
for  standard  \  sec.  cor.  marked  S.C.,  \  on  N.  face, 
and  raised  a  mound  of  stone  alongside. 

Pits,  impracticable. 

Top  of  ridge,  about  100  ft.  high. 

Ravine,  course  S.,  about  40  feet  deep. 


320 


PLANE   SURVEYING. 


80.00 


Set  a  post,  4-£-  ft.  long,  4  ins.   square,  with  marked 

stone,  12  ins.   in  the  ground,  for  standard  cor.  to 

sees.  32  and  33,  marked  : 

S.C.,  T.  13  N.,  R.  21  E.,  on  N. ; 
S.  33,  on  E.  ;  and 
S.  32,  on  W.  faces,  with  4  notches  on  E.  and  2  notches 

on  W.  faces,  and  raised  a  mound  of  stone  2  ft. 

high,  4^  ft.  base,  around  post. 
Land,  high  and  mountainous. 
Soil,  sandy,  gravelly,  and  rocky  ;  4th  rate. 
Timber,   pine,    and    fir,    80   chs.  ;    mostly   dead   and 

fallen  ;  some  thick  undergrowth,  same. 


375.  Specimen  Field  Notes  of  the  survey  of  Township  No. 
6  north,  Range  No.  34  east,  of  the  principal  base  and  meridian 
of  Montana  Territory. 


chains. 


16.40 
40.00 
79.96 


39.98 


79.96 


East,  on  random  line,  bet.  sees.  5  and  8. 

Va.  18°  45'  E. 
Over  rolling  ground. 
Road  to  Williamsburg,  course  S. 
Set  temporary  \  sec.  cor. 
Intersected  N.  and  S.  line  6  Iks.  N.  of  cor.  to  sees. 

4,  5,  8,  and  9. 
Thence  I  run 
N.  89°  56'  W.  on  true  line,  bet.  sees.  5  and  8.  with 

same  Va. 
Set  a  post  3  ft.  long,  3  ins.  square,  with  marked  stone, 

12  ins.  in  the  ground,  for  \  sec.  cor.  marked  ^  S. 

on  N.  face  ;  dug  pits,  18  x  18  x  12  ins.  E.  and  W. 

of  post  5£  ft.  dist.,  and  raised  a  mound  of  earth. 

1^  ft.  high,  31  ft.  base,  around  post. 
The  cor.  to  sees.  5,  6,  7,  and  8. 
Land,  rolling. 
Soil,  sandy  ;  2d  rate. 
No  timber. 


SUUVKY    OF   THE   PUBLIC    LANDS. 


321 


West,  on  random  line,  between  sees.  6  and  7. 

Over  rolling  ground. 

Road  to  Williamsburg,  course  S. 

Set  temporary  ^  sec.  cor. 

Intersect  west  boundary  of  township  15  Iks.  S.  of  cor. 

to  sees.  1,  6,  7,  and  12,  which  is  a  post,  4  ft.  long, 

4  ins.  square,  marked  : 
T.  6  N.S.  6  on  N.E. 
R.  34  E.S.  7  on  S.E. 
R.  33  E.S.  12  on  S.W., 

and  S.- 1  on  N.W.  faces,  with  pits,  18  x  18  x  12 

ins.  in  each  sec.,  5^  ft.  dist.,  and  mound  of  earth, 

2  ft.  high,  4£  ft.  base,  around  post. 
Thence  I  run 
S.  89°  54'  E.  on  a  true  line,  bet.  sees.  6  and  7,  with 

same  Va. 
Set  a  sandstone,  18  x  14  x  3  ins.,  12  ins.  in  the  ground, 

for  |-  sec.  cor.,  marked   \  on   N.   side;    dug  pits 

18  x  18  x  12  ins.  E.  and  W.  of  stone  5J  ft.  distant, 

and  raised  a  mound  of  earth,  1£  ft.  high,  3£  base, 

alongside. 

The  cor.  to  sees.  5,  6,  7,  and  8. 
Land,  rolling. 
Soil,  sandy  ;  2d  rate. 
No  timber. 


North,  on  a  random  line,  bet.  sees.  5  and  6. 
Va.  18°  45'  E. 

Over  rolling  ground. 

Set  temporary  \  sec.  cor. 

Intersect  N.  boundary  of  township  20  Iks.  E.  of  cor. 
to  sees.  5,  6,  31,  and  32,  which  is  a  sandstone 
30  x  12x6  ins.,  marked  with  5  notches  on  E.  and 
one  notch  on  W.  edges,  and  mound  of  stone,  2  ft. 
high,  4£  ft.  base,  alongside. 

Thence  I  run 


PLANE   SURVEYING. 


40.05 


80.05 


S.  0°  09'  E.  on  a  true  line  bet.  sees.  5  and  6,  with 
same  Va. 

Set  a  sandstone,  16  X  12  x  3  ins.  11  ins.  in  the  ground, 
for  ^  sec.  cor.  marked  \  on  W.  face ;  dug  pits, 
18  x  18  x  12  ins.,  N.  and  S.  of  stone,  5£  ft.  dist., 
and  raised  a  mound,  of  earth,  1^  ft.  high,  3^  ft. 
base,  alongside. 

The  cor.  to  sees.  5,  6,  7,  and  8. 

Land,  rolling. 

Soil,  sandy  ;  2d  rate. 

No  timber. 


INCLINATION  OF  THE  MERIDIAN.* 

376.    In  projecting  arcs  of  a  great  circle  it  is  of  the  utmost 
importance  that  the  surveyor  be  able  to  tell  the  inclination  of 
the  meridians  for  any  latitude,  and  for 
II  any  distance  of  eastings  or  westings. 

In  the  following  figure,  let  the  two 
arcs  AG  and  BG  be  two  arcs  of  a 
quadrant  of  the  meridian  1°  of  longi- 
tude apart.  Let  AB  =  the  arc  of  1° 
F  of  longitude  on  the  equator  =  69.16 
miles. 

Let  DE  be  an  arc  of  longitude  on 
any  parallel  of  latitude.     Also,  let  EH 
and  DH  be  the  tangents  of  those  me- 
ridians  meeting    in    the    earth's   axis 
produced,   and    corresponding    to    the 
parallel  of  latitude  DE. 
Then  the  line  EF=DF=cos  7,  =  cos  AD  or  BE.     Also, 
the  angle  DFE  =  1°,  and  the  angle  DHE  =  the  inclination  of 

*  These  articles  on  the  inclination  and  convergency  of  meridians,  and 
the  table  calculated  in  accordance  therewith,  are  substantially  those  given 
in  the  1886  catalogue  of  engineers'  and  surveyors'  instruments,  by  Buff 
and  Berger,  Boston,  Mass. 


SURVEY   OF    THE   PUBLIC   LANDS.  323 

the  meridians,  which  is  the  angle  we  wish  to  find,  aud  which 
we  will  represent  by  X°.  And  because  the  two  triangles  FDE 
and  DHE  are  on  the  same  base  ED,  and  isosceles,  their  vertical 
angles  vary  inversely  as  their  sides ;  and  we  have  the  equation, 
1°  x  EF=  X°  x  EH. 

But  EF  =  cos  L,  and  EH  =  cot  £ ; 

hence  X°  cot  L  =  1°  cos  L, 

or  X°  =  cos  L  -T-  cot  L  =  sin  L.  (a) 

That  is  to  say, 

The  inclination  of  the  meridians  for  any  difference  of  longitude 
varies  as  the  sine  of  the  latitude. 

Since  the  sine  of  the  latitude  is  the  inclination  in  decimals 
of  a  degree,  for  one  degree  of  longitude,  if  we  multiply  by  3600" 
we  shall  have  the  inclination  in  seconds  of  arc.  Then,  if  we 
divide  this  by  the  number  of  miles  in  one  degree  of  longitude 
on  that  latitude,  we  shall  have  the  inclination  due  to  one  mile 
on  that  parallel.  Thus,  for 

Latitude  43° log.  sin  =  9.833783 

Multiply  by  3600" "      «    =  3.556303 

3.390086 

Divide  by  50.66m.  =  1°  long,  on  that  L.  log.  =  1.704682 
48.46"  =  inclination  for  one  mile  of  long.  1.685404 

The  use  of  the  inclination,  as  found  by  the  preceding  article, 
is  to  show  the  surveyor  how  much  he  must  deflect  a  line  of 
survey  from  the  due  east  or  west,  to  have  it  meet  the  parallel 
at  a  given  distance  from  the  initial  point  of  the  survey ;  for 
it  will  be  remembered  that  a  parallel  of  latitude  is  a  curve 
having  the  cosine  of  the  latitude  for  its  radius.  And  the  line 
due  east  or  west  is  the  tangent  of  the  curve. 

Thus,  on  latitude  43°,  it  is  desired  to  project  a  six-mile  line 
west,  for  the  southerly  line  of  a  township. 

Remembering  that  in  an  isosceles  triangle  the  angle  at  the 
base  is  less  than  a  right  angle  by  half  the  angle  at  the  vertex, 
deflect  a  line  towards  the  pole  by  the  inclination  due  to  three 


H24  PLANE   SURVEYING. 

miles,  —  or  in  this  case  48.46"  x  3  =  2'.25"  ;  i.e.,  deflection  = 
\  inclination. 

The  table  on  next  page,  which  was  computed  from  the  for- 
mula (a)  above,  gives  the  inclination  for  one  mile,  and  for  six 
miles  on  any  parallel,  from  10°  to  60°  of  latitude;  also  the 
convergency  for  six  miles,  on  any  latitude. 

377.  The  Convergency  of  the  Meridian  is  readily  found  for 
any  given  distance  from  the  corresponding  inclination,  by  mul- 
tiplying the  sine  of  the  inclination  by  the  given  distance. 

Thus,  for  latitude  43°,  the  inclination  for  one  mile  is  48.46"; 
the  sine  of  which  is  0.000235.  This,  multiplied  by  the  number 
of  links  in  a  mile,  which  =  8,000,  we  have  the  convergency  for 
one  mile,  =  1.88  links. 

Multiplying  this  by  the  number  of  miles  in  a  township,  =  36, 
and  we  have  the  convergenc}*  for  a  township,  =  67.68  links. 
In  this  manner  were  the  convergeucies  of  the  Table  com- 
puted. 

378.  Deflection  of  Range-Lines  from  Meridian.    The  second 
column  of  the  table  shows  the  surveyor  how  much  he  must  de- 
flect the  range  lines  between  the  several  sections  of  a  township 
from  the  meridian,  in  order  to  make  the  consecutive  ranges 
of  sections  in  a  township  of  uniform  width,  for  the  purpose  of 
throwing   the  effects   of   convergency  into   the  most  westerly 
range  of  quarter-sections,  agreeably  to  law. 

Thus,  say  between  45°  and  55°  of  latitude,  the  inclination  is 
practically  1'  for  every  mile  of  easting  or  westing.  Then,  bear- 
ing in  mind  that  in  the  United  States  the  surveys  are  regarded 
as  projected  from  the  east  and  south  to  the  west  and  north,  the 
surveyor  must  project  the  first  range-line  between  the  sections 
of  a  township  in  those  latitudes  1'  to  the  left  of  the  meridian. 

The  second,  2' ;  the  third,  3' ;  and  so  on  to  the  fifth,  which 
must  be  5'  to  the  left  of  the  meridian  on  the  east  side  of  the 
township. 

By  this  means  all  the  convergency  of  the  township  is  thrown 
into  the  sixth,  or  westerly  range  of  sections,  as  the  law  directs. 


SURVEY   OF   THE   PUBLIC    LANDS. 


325 


The  fourth  column  of  the  table  below  shows  the  amount  of 
this  convergence-  This  column  is  also  useful  in  subdividing 
a  block  of  territory  embraced  by  two  standard  parallels  and  two 
guide  meridians  into  townships.  Thus,  starting  a  meridian 
from  a  standard  parallel  on  latitude  43°  N.,  for  the  western 
boundary  of  a  range  of  township,  —  say  the  first  one  west  from 
the  guide  meridian,  —  and  running  north,  say  four  townships, 
the  surveyor  must  make  a  point  that  is  east  of  the  six-mile  point 
on  the  northern  standard  parallel,  4  x  67.7  links  =  270. 8  links. 
The  second  meridian  should  fall  8  x  67.7  links  to  the  right  of 
the  twelve-mile  point. 

TABLE  OF  INCLINATION  AND  CONVERGENCY  OF  THE  MERIDIANS. 


13 

2 

Inclination  for 
one  mile. 

Inclination  for 
six  miles. 

Oonvergency  for 
one  township  of 
36  miles. 

•0 

0 

Inclination  for 
one  mile. 

Inclination  for 
six  miles. 

Convergency  for 
one  township  of 
36  miles. 

Latitude. 

Inclination  for 
one  mile. 

1  Inclination  for 
six  miles. 

Convergency  for 
one  township  of 
36  miles. 

o 

// 

/  // 

LINKS. 

•0 

// 

/  // 

LINKS. 

0 

/     // 

/   // 

LINKS. 

10 

9.18 

55 

13.0 

27 

26.52 

239 

36.9 

11 

50.19 

501 

70.1 

11 

10.13 

101 

14.2 

28 

27.66 

246 

38.6 

46 

52.00 

512 

72.6 

12 

11.07 

106 

15.5 

20 

28.85 

253 

40.2 

46 

53.83 

523 

75.2 

13 

12.02 

112 

16.8 

80 

30.03 

303 

41.9 

47 

55.67 

534 

77.8 

14 

12.98 

118 

18.1 

31 

31.26 

307 

43.6 

48 

57.67 

546 

80.6 

15 

13.96 

124 

19.4 

82 

32.49 

315 

45.4 

49 

59.83 

559 

83.5 

16 

14.93 

130 

20.7 

88 

33.83 

323 

47.2 

50 

1  02.00 

612 

86.5 

17 

15.92 

136 

22.0 

34 

35.17 

331 

49.1 

61 

1  04.17 

625 

89.7 

18 

16.91 

141 

23.4 

86 

36.50 

339 

50.9 

62 

1  06.67 

640 

93.0 

19 

17.93 

147 

24.9 

86 

37.83 

346 

62.7 

68 

1  09.17 

665 

96.4 

20 

18.94 

164 

26.5 

37 

39.17 

355 

54.7 

64 

1  16.67 

710 

100.0 

21 

19.98 

200 

27.8 

38 

40.67 

404 

56.8 

66 

1  14.33 

726 

103.7 

22 

21.02 

206 

29.3 

39 

42.17 

413 

58.8 

56 

1  17.17 

743 

107.6 

23 

22.10 

213 

30.8 

40 

43.67 

422 

60.9 

57 

1  20.00 

800 

111.8 

24 

2.3.17 

219 

32.3 

41 

45.17 

431 

63.1 

58 

1  22.00 

819 

116.2 

25 

24.30 

226 

33.8 

42 

46.85 

441 

65.4 

59 

1  26.66 

840 

120.9 

26 

25.38 

232 

35.4 

1:5 

48.52 

451 

07.7 

60 

130.00 

900 

125.7 

For  details  of  instruction  in  United  States  Government  Surveying,  see 
Iliiwcs'  System  of  "Rectangular  Surveying,"  Kurt's  "Key  to  Solar  Com 
pass,"  and  Clevenger's  "Government  Surveying." 


CHAPTER  VII. 


CITY  SUKVEYING. 


INTRODUCTION. 

379.  In  the  broadest  sense,  the  duties  of  a  city  engineer  in 
a  large  city  are  many  and  varied.  His  knowledge  and  judg- 
ment are  required  in  the  location  of  the  city,  the  laying  out  of 
streets,  and  the  fixing  of  suitable  grades  therefor,  the  establish- 
ment of  a  proper  water  supply,  the  designing  of  a  suitable  sys- 
tem of  sewers,  the  improvement  of  the  waterways,  and  the 
planning  of  necessary  bridges  and  buildings.  Following  his 
judicial  functions  as  a  designer  are  his  ministerial  functions  as 
a  constructor.  The  field  which  is  thus  opened  before  him,  in 
carrying  into  execution  the  plans  for  the  various  public  works, 
is  a  very  wide  one. 

As  the  borough  grows  and  expands  into  the  metropolis,  its 
needs  in  the  directions  mentioned  increase  until  a  division  of 
labor  and  responsibility  becomes  expedient  and  necessary.  In 
securing  the  best  results  in  engineering  practice,  as  in  other 
work,  the  tendency  is  towards  specialties  ;  so  that  in  many  cities, 
in  order  to  secure  the  services  of  the  best  men,  and  also  the 
best  results,  the  numerous  and  important  duties  connected  with 
city  engineering  have  been  separated.  The  province  of  this 
work,  which  is  not  a  treatise  on  engineering,  but  on  land  sur- 
veying, makes  it  proper  to  treat  in  this  chapter,  as  thoroughly 
as  the  intention  and  limits  of  the  work  allow,  only  what  may 
be  classed  under  the  head  of  surveying,  whether  it  be  performed 
as  the  special  work  of  the  city  or  town  surveyor,  or  as  among 
the  duties  of  the  city  engineer,  —  the  qualifications  of  the 


CITY    SURVEYING.  327 

former  by  no  means  fitting  a  man  to  perform  the  varied  duties 
of  the  latter. 

Although  this  work  is  intended  for  the  instruction  of  the 
student,  not  of  the  experienced  surveyor,  and  hence  in  many 
things  may  go  into  details  which  to  the  latter  may  seem  unim- 
portant, it  is  impossible  in  the  limits  of  a  chapter  to  impart  a 
thorough  knowledge  of  the  duties  of  a  citv  or  town  surveyor,  — 
indeed,  even  to  mention  all  his  duties  and  the  many  operations 
and  methods  which  only  a  long  and  varied  practice  can  impart. 
General  methods  will  be  given  and  discussed,  but  any  survevor 
of  a  practical  turn  of  mind  will  have  his  own  methods  of  per- 
forming much  of  the  routine  work  pertaining  to  his  situation. 

It  is  not  in  harmony  with  the  plan  of  this  work  to  go  into  the 
statement  in  this  chapter  of  any  elaborate  theories  regarding 
surveying  and  the  instruments  used  therein,  but  to  endeavor  to 
give  some  methods  which  are  found  to  be  applicable  in  practice 
and  to  give  good  practical  results.  A  thorough  knowledge  of  any 
one  good  method  of  performing  a  certain  work  is  of  much  more 
value  to  the  student  than  a  misty  idea  of  numerous  methods. 

Under  the  two  leading  heads  of  this  chapter,  field  instruments 
and  work  and  office  instruments  and  work,  theoretical  discus- 
sions will  not  be  entered  into ;  not  because  they  do  not  possess 
much  value,  but  because  we  conceive  that  they  are  not  adapted 
to  the  student's  present  needs  and  most  rapid  advancement. 
Under  the  former  head,  in  the  light  of  the  work  which  is  likely 
to  engage  the  greater  part  of  the  surveyor's  time,  field  instru- 
ments and  methods  of  using  them  will  be  described.  Under 
the  latter,  the  nature  of  office  plans  and  records  will  be  de- 
scribed, the  instruments  and  methods  used  in  the  work  of  pro- 
ducing the  plans  having  been  described  in  other  chapters. 

In  dividing  land  and  locating  the  boundaries  between  parties 
it  is  evident  that  the  greater  the  value  or  the  prospective  value 
of  said  lands,  the  more  delicate  should  be  the  instruments,  and 
the  more  exact  the  methods  used  in  the  work.  The  methods 
and  instruments  which  would  for  all  practical  purposes  be  suffi- 
ciently exact  for  the  location  of  a  line  fence  iu  the  country, 


328  PLANE   SURVEYING. 

where  land  might  be  purchased  for  Si 00  per  acre,  would  not  at 
all  meet  the  requirements  in  locating  in  a  city  a  line  between  two 
parties  on  land  worth  $100  per  front  foot.  This  fact  becomes 
the  more  evident  when  we  consider  that  the  structures  placed 
upon  party  lines  in  a  city  are  so  much  more  substantial  and  per- 
manent in  their  nature  than  those  thus  located  in  the  country. 
To  meet  these  considerations  we  shall  find  that  while  some  of 
the  methods  of  land  surveying  previously  described  in  this  work, 
and  the  instruments  used  therein,  are  applicable  to  the  purposes 
of  city  surveying,  many  of  the  methods  will  be  more  exact,  and 
the  instruments  more  numerous  and  delicate. 

Following  the  plan  heretofore  pursued  in  this  work,  we  will, 
before  discussing  the  work  of  the  city  surveyor,  describe  the 
instruments  (not  described  in  previous  chapters)  of  most  gen- 
eral use  in  his  work,  and  explain  their  adjustments  and  the 
general  methods  of  using  them.  These  instruments  are  the 
transit  and  rods,  steel  tapes,  measuring-rods,  pocket-thermome- 
ter, hand-level,  spring-balance,  plummet,  Y-level,  le veiling-rods, 
and  rod-levels. 


SECTION   I. 

INSTRUMENTS,  THEIR  ADJUSTMENTS  AND  GENERAL  USES. 

A.    FIELD   INSTRUMENTS. 

1 

380.  The  Transit.     Full  description  of  the  transit,  its  adjust- 
ment and  uses,  may  be  found  in  Chapter  II. 

381.  As  precision  is  the  distinguishing  feature  of  city  and 
town  surveying,  the  magnetic  needle,  which  is  usually  found 
upon  the  transits,  is  in  this  work  of  but  little  use.     Angles  in 
carefully  made  surveys  are  now  taken  on  the  horizontal  gradu- 
ated circle  of  the  transit.    The  instructions  already  given  in  this 
work  regarding  the  magnetic  needle  are  sufficient  reason  for  the 


TRANSIT, 

WITH  GRADIENTER,  LEVEL  TO  TELESCOPE,  AND  VERTICAL  ARC,  AS  MADE  BV 
YOUNG  &  SONS,  PHILADELPHIA,  PA. 


FIELD    INSTRUMENTS.  331 

above.  It  is,  however,  desirable  that  in  each  city  and  town 
the  true  meridian  should  be  determined  and  permanently  marked. 
Besides  being  useful  in  many  other  ways  which  will  suggest 
themselves,  it  will  be  of  great  use  as  an  aid  in  determining  the 
situation  of  lines  described  by  their  bearings  in  old  deeds,  the 
date  of  the  old  survey  being  known. 

382.  The  stadia-hairs  *  and  vertical  circle  for  stadia-measure- 
ments are  useful  attachments,  and  the  telescope  should  by  all 
means  have  a  long  level-tube  attached,  as  this  is  of  much  use 
in  city  and  town  work  in  running  grade  lines  and  in  levelling 
for  short  distances.  .  After  the  level  and  the  manner  of  using  it 
have  been  described,  the  operation  of  running  a  grade  line  will 
be  explained. 

383.  Rods.  Besides  the  usual  iron-pointed  wooden  rods,  very 
convenient  rods,  or  pickets,  for  use  with  the  transit,  may  be  made 
of  gas-pipe  about  three-quarters  of  an  inch  in  diameter  drawn 
out  on  one  end  to  a  point,  and  painted  in  alternate  sections  of 
red  and  white,  —  red  preferred  to  black  because  against  red  the 
cross-hairs  can  be  seen. 

384.  It  is  b}-  no  means  as  easy  a  matter  to  run  a  straight 
line  with  a  transit  as  at  first  thought  it  may  seem  to  the  student. 
After  the  selection  of  suitable  weather,  reversing  at  every  ex- 
tension, care  in  handling  the  instrument,  and  with  a  correspond- 
ing degree  of  care  on  the  part  of  assistants,  the  results  are  not 
always  what  the  most  careful  would  desire. 

385.  In  marking  a  line  with  stakes,  it  is  convenient  to  have 
stake-wood  which,  in  cross-section,  has  one  dimension  greater 
than  the  other.     If,  in  setting  the  stake,  it  always  be  placed 
with  its  broader  side  towards  the  instrument,  its  position  will 
afterwards  tell  one  at  a  glance  in  which  direction  the  line  was 
run.     This  is  important  when  several  stakes  are  set  on  different 

*  See  Articles  148  to  152,  Stadia  Measurements. 


332  PLANE   SURVEYING. 

lines  near  their  intersection,  as  it  will  often  be  the  means  of 
avoiding  confusion  and  the  resulting  errors. 

386.  Steel  Tapes,  etc.     Before  making  an}-  important  meas- 
urements for  a  city  or  town,  it  is  necessary,  in  order  to  avoid 
subsequent  confusion,  that  a  standard  of  measurement  should 
be  adopted.     In  many  parts  of  an  old  city  or  town  the  intro- 
duction of  a  new  standard  would  bring  inextricable  confusion. 
If  there  be  a  standard,  even  though  it  has  not  been  carefully 
preserved,  it  should,  if  possible,  be  ascertained  and  regarded. 
When,  however,  it  is  at  the  option  of  the  surveyor  to  select  his 
standard,  the  United  States  standard  should,  as  tending  to  uni- 
formity, be  adopted  in  this  country.     Standard  rods  may  be 
procured  of  the  government.     With  these  rods  tape  lines  and 
other  instruments  used  for  a  line  purpose  should  be  compared, 
and  the  variation  noted.     It  is  desirable,  also,  for  purposes  of 
comparison,  that  a  standard,  50  feet  or  100  feet,  at  a  known 
temperature,  should  be  carefully  laid  down  with  these  rods  in 
the  corridor  of   some  building,  or  in  some  other  convenient 
place. 

Very  accurate  measuring  may  be  done  with  graduated  wooden 
rods  properly  shod  with  metal  ends.  These  rods  are  necessarily  of 
but  moderate  length  ;  hence,  work  with  them  is  correspondingly 
slow.  For  city  work,  steel  tapes  are  now  in  very  general  use  ; 
and,  when  properly  handled,  give  very  satisfactory  results.  They 
are  of  different  lengths  and  of  different  widths.  For  measur- 
ing full  hundreds  over  tolerably  level  ground  the  narrow  tape, 
-^  inch  wide  and  200  feet  long,  is  very  convenient.  For  general 
city  use  the  100-feet  tape,  f  inch  in  width,  is  most  convenient. 

387.  As  a  rule  measurements  will  be  made  with  the  tape  in 
a  horizontal  position.     If  not  so  held,  the  measurements  will 
afterwai'ds  be  reduced  to  the  horizontal.     In  order  to  determine 
the  horizontal,  a  hand-level  is  used  to  ascertain  the  difference 
in  elevation  of  the  ground  at  the  two  ends  of  the  tape.     A  cut 
and  description  of  this  convenient  little  instrument  is  given 
below. 


FIELD    INSTRUMENTS.  333 

Locke's  Hand-Level  consists  of  a  brass  tube  about  6  inches 
long,  having,  as  shown  in  the  figure,  a  small  level  on  top  and 
near  the  object  end,  there  being  also  an  opening  in  the  tube 
beneath,  through  which  the  bubble  can  be  seen,  as  reflected  by 
a  glass  prism,  immediately  under  the  level.  Both  ends  of  the 
tube  are  closed  by  plain  glass  settings  to  exclude  the  dust,  and 
there  is  at  the  inner  end  of  the  sliding  or  eye  tube  a  semicircu- 
lar convex  lens,  which  serves  to  magnify  the  level  bubble,  and 
cross-wire  underneath,  while  it  allows  the  object  to  be  clearly 
seen  through  the  open  half  of  the  tube. 


The  cross-wire  is  fastened  to  a  little  frame  moving  under  the 
level-tube,  and  adjusted  to  its  place  by  the  small  screw  shown 
on  the  end  of  the  level-case.  The  level  of  any  object  in  line 
with  the  eye  of  the  observer  is  determined  by  sighting  upon  it 
through  the  tube,  and  bringing  the  air-bubble  of  the  level  into 
a  position  where  it  is  bisected  by  the  cross-wire. 

A  short  telescope  is  sometimes  applied  in  place  of  the  plain 
glass  ends,  enabling  levels  to  be  taken  at  greater  distances  and 
with  increased  accuracy. 

If  one  or  both  ends  of  the  tape  be  held  up,  the  point  on  the 
ground  vertically  under  the  end  of  the  tape  will  be  determined 
by  means  of  the  plummet,  which  here  needs  no  description 
further  than  to  sav  that  its  sides  should  make  such  an  angle 
with  each  other  as  not  to  prevent  the  observer  when  using  it 
from  seeing  its  point ;  neither  should  it  be  so  long  as  to  be 
unsteady. 

In  all  extended  and  important  measurements  regard  must  be 
had  in  using  the  steel  tape  to  standard,  temperature,  sag,  and 
wind . 

Before  using  a  tape  its  relation  to  the  standard  should  be 


334  PLANE   SURVEYING. 

determined  by  comparison  with  the  standard,  marked  as  pre- 
viously described,  and  the  variation  noted. 

388.  All  important  measurements,  no  matter  at  what  tem- 
perature made,  should  be  reduced  to  a  standard  temperature ; 
for  if,  at  a  certain  temperature,  we  determined  with  a  steel  taue 
the  distance  apart  of  two  points,  at  a  higher  temperature  that 
distance  on  the  same  tape  would  be  less  because  the  tape  is 
longer ;  or,  at  a  lower  temperature,  greater,  because  the  tape  is 
shorter.     The  temperature  of  the  air  at  the  time  of  measurement 
is  ascertained  by  means  of  a  small  thermometer  which  can  be 
exposed  with  the  tape,  and  which  is  so  protected  that,  when  not 
in  use,  it  can  be  safely  carried  in  the  pocket.     The  standard 
temperature  to  which  all  measurements  should  be  reduced  may 
be  taken  at  pleasure.     The  correction  for  expansion  and  con- 
traction of  the  steel  tape  by  heat  and  cold  is  0.000006  per  unit 
per  degree  F. 

389.  When  the  tape  is  held  suspended,  it  will  always  sag  in 
a  vertical  direction.      Hence    the  horizontal  distance  between 
the  extreme  graduations  will  be  less  than  if  there  were  no  sag. 
For  this  reason,  when  used  to  measure  the  distance  between 
two  points,  it  will,  without  correction,  give  a  result  too  great; 
when  used  without  correction  to  lay  down  a  given  distance,  it 
will  give  it  too  small.     While  a  formula  may  be  derived  by 
which  to  make  a  correction  for  sag,  it  will  be  found  quite  as 
satisfactory  to  determine  it  by   actual  trial.     The   amount  of 
sag  will  of  course  depend  upon  the  tension,  or  pull.     This  may 
be  regulated  by  using  at  one  end  of  the  tape  a  small  spring- 
balance.     It  is,  however,  very  desirable  that  on  important  work 
the  same  men  at  the  same  ends  of  the  tape  should  make  all 
measurements.     The    experience    gained  in  working  together 
will  be  a  most  important  factor  in  securing  uniform  results. 

The  effect  of  wind  is  in  the  same  direction  as  that  of  sag. 
While  much  of  the  work  of  the  surveyor,  particularly  that  in- 
volving short  measurements,  must  be  done  regardless  of  wind, 


FIELD   INSTRUMENTS.  335 

no  good  results  in  long  and  important  measurements  can  be 
secured  in  windy  weather.  The  best  correction  for  wind  is  to 
wait  for  a  calm.  In  windy  weather  a  narrow  tape,  as  it  ex- 
poses less  surface  to  the  wind,  is  useful. 

390.  To  illustrate  what  has  been  said  in  regard  to  the  correc- 
tions to  be  applied  to  measurements  made  with  the  steel  tape, 
let  us  suppose  two  examples. 

First.  With  a  steel  tape  100  feet  long  (f  inch  wide)  sus- 
pended each  length  at  one  or  both  ends,  the  temperature  of 
the  air  being  79°  F.,  the  distance  on  the  tape  between  two 
points  is  found  to,  be  550  feet  6|  inches.  If  the  tape  is  •£ 
inch  longer  than  the  standard,  and  parts  of  its  length  propor- 
tionately longer,  the  standard  temperature,  60°  F.,  and  the  sag 
£  inch  in  100  feet,  what  are  the  corrections,  and  what  is  the 
actual  distance  between  the  points  ? 

On  account  of  differing  from  the  standard,  as  the  tape  is  too 
long,  the  distance  obtained  is  too  short ;  the  correction  for 
standard  is  therefore  additive.  On  account  of  difference  in 
temperature,  the  temperature  being  higher  than  the  standard, 
as  the  tape  is  too  long,  the  distance  obtained  is  too  short; 
the  correction  for  temperature  is  therefore  additive.  On 
account  of  the  sag,  as  the  tape  is  thereby  made  too  short,  the 
distance  obtained  is  too  long ;  the  correction  for  sag  is  there- 
fore subtractive. 

Correction  for  standard : 

|  in.  x  5£  =  |£  in.  additive. 
Correction  for  temperature  (79°  —  60°  =  19°)  : 

0.000006  ft.  x  550  x  19  =  0.0627  ft. 

0.0627  ft.  x  12  =  0.7524  in.  =  |f  in.  additive. 

Correction  for  sag : 

£  in.  x  5£  =  |f  in.  subtractive. 
Total  correction : 

+  M.  in.  _f_  i£  in.  _  II  in.  =  +  T^  in.  additive. 


336  PLANE   SUE V EYING. 

Actual  distance  between  points  : 

550  ft.  6J  in.  +  fa  in.  =  550  ft.  6ff  in. 

Second.  Suppose  it  be  required,  —  other  things  being  as 
before, — to  locate  with  the  steel  tape,  when  the  temperature 
of  the  air  is  52°  F.,  two  points  which  shall  at  the  standard 
temperature  be  225  feet  4£  inches  apart. 

What  length  on  the  tape  must  be  taken  ? 

Correction  for  standard : 

{•  in.  x  2^  =  -£2  m-  subtractive. 
Correction  for  temperature  (60°  —  52°  =  8°)  : 

0.000006  ft.  x  225  x  8  =  0.0108  ft. 

0.0108  ft,  X  12  =  0.1296  in.  =  fa  in.  additive. 

Correction  for  sag : 

\  in.  x  2£  =  ^|  in.  additive. 
Total  correction : 

—  fa  in.  -f  u42  in-  +  it  in-  =  +if  in-  additive. 
Length  to  be  taken  on  tape  : 

225  ft.  4|  in.  +  if  in.  =  225  ft.  4ff  in. 

When  the  tape  is  not  suspended,  correction  for  sag  will  not 
be  made. 

In  short  and  less  important  measurements  the  same  attention 
to  corrections  is  not  necessary. 

In  practice,  the  above  method  has  been  found  to  give  satis- 
factory results. 

391.  In  placing  stakes  to  hold  measurements,  it  is  best,  and 
in  harmony  with  the  method  suggested  for  placing  them  on 
instrument  lines,  to  set  them  with  the  greater  dimension  of 
cross-section  in  the  direction  in  which  the  measurement  is 
being  made. 

Measuring  is  a  very  important  part  of  the  work  of  the  sur- 
veyor. Even  when  done  with  the  greatest  care,  it  is  difficult  to 
obtain  results  entirely  satisfactory. 


FIELD   INSTRUMENTS.  337 

Measurements  which  are  to  be  directly  compared,  or  are  to 
be  used  in  connection,  as  in  locating  parallel  lines,  should  be 
made  under  circumstances  as  nearly  as  possible  identical. 
Experience  and  a  correct  idea  of  the  importance  of  the  work 
will  enable  the  surveyor  to  determine  the  degree  of  accuracy 
therein  necessary. 

LEV  ELLING-lNSTRUMENTS . 

392.  The  Y-LeveL    Of  the  different  varieties  of  the  levelling- 
instrument,  that  termed  the  Y-level  has  been  almost  Universally 
preferred  by  American  engineers,  on  account  of  the  facility  of 
its  adjustment  and  superior  accurac}*. 

The  engraving  represents  a  twenty-inch  Y-level  as  made  by 
W.  and  L.  E.  Gurley,  Troy,  N.Y. 

393.  The  Telescope  has  at  each  end  a  ring  of  bell-metal, 
turned  very  truly,  and  both  of  exactly  the  same  diameter ;  by 
these  it  revolves  in  the  wyes,  or  can  be  at  pleasure  clamped  in 
any  position  when  the  clips  of  the  wyes  are  brought  down  upon 
the  rings,  by  pushing  in  the  tapering-pins. 

394.  The  Level  or  ground  bubble  tube  is  attached  to  the 
under  side  of  the  telescope,  and  furnished  at  the  different  ends 
with   the   usual   movements,  in   both   horizontal   and    vertical 
directions. 

The  aperture  of  the  tube,  through  which  the  glass  vial 
appears,  is  about  5£  inches  long,  being  crossed  at  the  centre 
by  a  small  rib  or  bridge,  which  greatly  strengthens  the  tube. 

The  level-scale  which  extends  over  the  whole  length  is 
graduated  into  tenths  of  an  inch,  and  figured  at  every  fifth 
division,  counting  from  zero  at  the  centre  of  the  bridge  ;  the 
scale  is  set  close  to  the  glass. 

The  bubble  vial  is  made  of  thick  glass  tube,  selected  so  as  to 
have  an  even  bore  from  end  to  end,  and  finely  ground  on  its 
upper  interior  surface,  that  the  run  of  the  air-bubble  may  be 
uniform  throughout  its  whole  range. 


340  PLANE   SURVEYING. 

395.  The  Wyes  are  made  large  and  strong,  of  the  best  bell- 
metal,  and  each  has  two  nuts,  both  being  adjustable  with  the 
ordinary  steel  pin. 

The  clips  are  brought  down  on  the  rings  of  the  telescope- 
tube  by  the  Y-pins,  which  are  made  tapering,  so  as  to  clamp 
the  rings  very  firmly. 

The  clip  of  one  of  the  wyes  has  a  little  pin  projecting  from 
it,  which,  entering  a  recess  filed  in  the  edge  of  the  ring,  insures 
the  vertical  position  of  the  level  and  cross-wire. 

396.  The  Level-Bar  is  made  round,  of  the  best  bell-metal, 
and  shaped  so  as  to  possess  the  greatest  strength  in  the  parts 
most  STibject  to  sudden  strains. 

Connected  with  the  level-bar  is  the  head  of  the  tripod- 
socket. 

397.  The  Tripod-Socket  is  compound ;   the   interior  spindle 
Z>,  sectional  view,  upon  which  the  whole  instrument   is   sup- 
ported,  is  made  of   steel,  and  nicely  ground,  so  as  to  turn 
evenlv  and  firmly  in   a  hollow  cylinder   of    bell-metal ;    this, 
again,  has  its  exterior  surface  fitted  and  ground  to  the  main 
socket  EE  of  the  tripod-head. 

The  bronze  cylinder  is  held  upon  the  spindle  by  a  washer 
and  screw,  the  head  of  the  last  having  a  hole  in  its  centre, 
through  which  the  string  of  the  plumb-bob  is  passed. 

THE  ADJUSTMENTS. 

398.  The  three  adjustments  of  the  level  which  the  surveyor 
usually  has  to  attend  to  are  the  following ; 

1.  To  adjust  the  line  of  collimation,  or,  in  other  words,  to 
bring  both  wires  into  the  optical  axis,  so  that  their  point  of 
intersection  will  remain  on  any  given  point   during  an  entire 
revolution  of  the  telescope. 

2.  To  bring  the  level-bubble  parallel  with  the  bearings  of  the 
V-rings,  and  with  the  longitudinal  axis  of  the  telescope. 


FIELD    INSTRUMENTS.  341 

3.  To  adjust  the  wyes,  or  to  bring  the  bubble  into  a  position 
at  right  angles  to  the  vertical  axis  of  the  instrument. 

399.  To  Adjust  the  Line  of  Collimation,  set  the  tripod  firmly, 
remove  the  Y-pins  from  the  clips,  so  as  to  allow  the  telescope 
to  turn  freely,  clamp  the  instrument  to  the  tripod-head,  and,  by 
the  levelling  and  tangent  screws,  bring  either  of  the  wires  upon 
a  clearly  marked  edge  of  some  object,  distant  from  100  to  500 
feet. 

.  Then,  with  the  hand,  carefully  turn  the  telescope  half-way 
around,  so  that  the  same  wire  is  compared  with  the  object 
assumed. 

Should  it  be  found  above  or  below,  bring  it  half-way  back  by 
moving  the  capstan-head  screws  at  right  angles  to  it,  remem- 
bering always  the  inverting  property  of  the  eye-piece ;  now 
bring  the  wire  again  upon  the  object,  and  repeat  the  first 
operation,  until  it  will  reverse  correctlv. 

Proceed  in  the  same  manner  with  the  other  wire  until  the 
adjustment  is  completed. 

Should  both  wires  be  much  out,  it  will  be  well  to  bring  them 
nearly  correct  before  either  is  entirely  adjusted. 

When  this  is  effected,  unscrew  the  covering  of  the  eye-piece 
centring-screws,  shown  in  the  sectional  view  at  AA,  and  move 
each  pair  in  succession  with  a  small  screw-driver,  until  the  wires 
are  brought  into  the  centre  of  the  field  of  view. 

The  inverting  property  of  the  eye-piece  does  not  affect  this 
operation,  and  the  screws  are  moved  direct. 

To  test  the  correctness  of  the  centring,  revolve  the  telescope, 
and  observe  whether  it  appears  to  shift  the  position  of  an 
object. 

Should  any  movement  be  perceived,  the  centring  is  not 
perfectly  effected. 

It  may  here  be  repeated,  that  in  all  telescopes  the  position 
and  adjustment  of  the  line  of  collimation  depends  upon  that  of 
the  object-glass  ;  and,  therefore,  that  the  movement  of  the  eye- 
piece does  not  affect  the  adjustment  of  the  wires  in  any  respect. 


342  PLANE   SURVEYING. 

When  the  centring  has  been  once  effected,  it  remains  per- 
manent, the  cover  being  screwed  on  again  to  conceal  and 
protect  it  from  derangement  at  the  hands  of  the  curious  or 
inexperienced  operator. 

400.  To  Adjust  the  Level-Bubble.  Clamp  the  instrument 
over  either  pair  of  le veiling-screws,  and  bring  the  bubble  into  the 
centre  of  the  tube. 

Now  turn  the  telescope  in  the  wyes,  so  as  to  bring  the  level- 
tube  on  either  side  of  the  centre  of  the  bar.  Should  the  bubble 
run  to  the  end,  it  would  show  that  the  vertical  plane  passing 
through  the  centre  of  the  bubble  was  not  parallel  to  that  drawn 
through  the  axis  of  the  telescope-rings. 

To  correct  the  error,  bring  the  bubble  entirely  back,  with 
the  capstan-head  screws,  which  are  set  in  either  side  of  the 
level-holder,  placed  usually  at  the  object  end  of  the  tube. 

Again  bring  the  level-tube  over  the  centre  of  the  bar,  and 
the  bubble  to  the  centre ;  turn  the  level  to  either  side,  and,  if 
necessary,  repeat  the  correction  until  the  bubble  will  keep  its 
position,  when  the  tube  is  turned  half  an  inch  or  more  to  either 
side  of  the  centre  of  the  bar. 

The  necessity  for  this  operation  arises  from  the  fact  that 
when  the  telescope  is  reversed  end  for  end  in  the  wyes  in  the 
other  and  principal  adjustment  of  the  bubble,  we  are  not  certain 
of  placing  the  level-tube  in  the  same  vertical  plane  ;  and  there- 
fore it  would  be  almost  impossible  to  effect  the  adjustment 
without  a  lateral  correction. 

Having  now,  in  great  measure,  removed  the  preparatory 
difficulties,  we  proceed  to  make  the  level-tube  parallel  with 
the  bearings  of  the  Y-rings. 

To  do  this,  bring  the  bubble  into  the  centre  with  the  levelling- 
screws,  and  then,  without  jarring  the  instrument,  take  the 
telescope  out  of  the  wyes  and  reverse  it  end  for  end.  Should 
the  bubble  run  to  either  end,  lower  that  end,  or,  what  is  equiva- 
lent, raise  the  other  by  turning  the  small  adjusting-nuts,  on  one 
end  of  the  level,  until  by  estimation  half  the  correction  is  made  ; 


FIELD   INSTRUMENTS.  343 

again  bring  the  bubble  into  the  centre,  and  repeat  the  whole 
operation,  until  the  reversion  can  be  made  without  causing  any 
change  in  the  bubble. 

It  would  be  well  to  test  the  lateral  adjustment,  and  make 
such  correction  as  may  be  necessary  in  that,  before  the  hori- 
zontal adjustment  is  entirely  completed. 

401.  To  Adjust  the  Wyes.  Having  effected  the  previous 
adjustments,  it  remains  now  to  describe  that  of  the  wyes,  or, 
more  precisely,  that  which  brings  the  level  into  position  at 
right  angles  to  the  vertical  axis,  so  that  the  bubble  will  remain 
in  the  centre  during1  an  entire  revolution  of  the  instrument. 

To  do  this,  bring  the  level-tube  directly  over  the  centre  of 
the  bar,  and  clamp  the  telescope  firmly  in  the  wyes,  placing 
it,  as  before,  over  two  of  the  levelling-screws,  unclamp  the 
socket,  level  the  bubble,  and  turn  the  instrument  half-way 
around,  so  that  the  level-bar  may  occupy  the  same  position 
with  respect  to  the  levelling-screws  beneath. 

Should  the  bubble  run  to  either  end,  bring  it  half-way  back 
by  the  Y-nuts  on  either  end  of  the  bar ;  now  move  the  telescope 
over  the  other  set  of  levelling-screws,  bring  the  bubble  again 
into  the  centre,  and  proceed  precisely  as  above  described, 
changing  to  each  pair  of  screws,  successively,  until  the  adjust- 
ment is  very  nearly  perfected,  when  it  may  be  completed  over 
a  single  pair. 

The  object  of  this  approximate  adjustment  is  to  bring  the 
upper  parallel  plate  of  the  tripod-head  into  a  position  as  nearly 
horizontal  as  possible,  in  order  that  no  essential  error  may 
arise,  in  case  the  level,  when  reversed,  is  not  brought  precisely 
to  its  former  situation.  When  the  level  has  been  thus  com- 
pletely adjusted,  if  the  instrument  is  properly  made,  and  the 
sockets  well  fitted  to  each  other  and  the  tripod-head,  the  bubble 
will  reverse  over  each  pair  of  screws  in  any  position. 

Should  the  surveyor  be  unable  to  make  it  perform  correctly, 
he  should  examine  the  outside  socket  carefully  to  see  that  it 
sets  securely  in  the  main  socket,  and  also  notice  that  the  clamp 
does  not  bear  upon  the  ring  which  it  encircles. 


344  PLANE    SURVEYING. 

When  these  are  correct,  and  the  error  is  still  manifested,  it 
will  probably  be  in  the  imperfection  of  the  interior  spindle. 

After  the  adjustments  of  the  level  have  been  effected,  and 
the  bubble  remains  in  the  centre,  in  any  position  of  the  socket, 
the  surveyor  should  turn  the  telescope  in  the  wyes  until  the  pin 
on  the  clip  of  the  wve  will  enter  the  little  recess  in  the  ring  to 
which  it  is  fitted,  and  by  which  is  insured  the  vertical  position 
of  the  spirit-level  and  cross-wire. 

When  the  pin  is  in  its  place,  the  vertical  wire  may  be  applied 
to  the  edge  of  a  building ;  and  in  case  it  should  not  be  parallel 
with  it,  two  of  the  cross-wire  screws  that  are  at  right  angles  to 
each  other  may  be  loosened,  and  by  the  screws  outside,  the 
cross-wire  ring  turned  until  the  wire  is  vertical ;  the  line  of  col- 
limation  must  then  be  corrected  again  and  the  adjustments  of 
the  level  will  be  complete. 

402.  To  Use  the  Level  Set  the  legs  firmly  into  the  ground. 
The  bubble  should  then  be  brought  over  each  pan-  of  levelling- 
screws  successively  and  levelled  in  each  position,  any  correction 
that  may  appear  necessary  being  made  in  the  adjustments. 

Bring  the  wires  precisely  in  focus  and  the  object  distinctly  in 
view,  so  that  all  errors  of  parallax  may  be  avoided. 

This  error  is  seen  when  the  eye  of  an  observer  is  moved  to 
either  side  of  the  centre  of  the  eye-piece  of  a  telescope,  in  which 
the  foci  of  the  object  and  eye-glasses  are  not  brought  precisely 
upon  the  cross-wires  and  object ;  in  such  a  case  the  wires  will 
appear  to  move  over  the  surface,  and  the  observation  will  be 
liable  to  inaccuracy. 

In  all  instances  the  wires  and  object  should  be  brought  into 
view  so  perfectly  that  the  cross-wires  will  appear  to  be  fastened 
to  the  surface,  and  will  remain  in  that  position  however  the  e%ye 
is  moved. 

Care  should  be  exercised  during  an  observation,  last  the  hand 
touching  the  instrument  inadvertently,  or  a  foot  placed  near 
the  leg  of  the  tripod,  impair  the  adjustment. 

The  weight  of  a  level  having  a  20-inch  telescope,  with  level- 


NEW  YORK.  P 

LEVELLING-RODS. 


FIELD  INSTRUMENTS.  347 

ling-head,  exclusive  of  the  tripod,  is  between  thirteen  and  four- 
teen pounds. 

IiEVELLING-RODS . 

403.  The  various  levelling-rods  used  by  American  engineers 
are  made  in  two  or  more  parts,  which  slide  from  each  other  as 
they  are  extended  in  use. 

404.  The  New  York  Rod.     This  rod,  which  is  shown  in  the 
engraving  as  cut  in  two,  so  that  the  ends  may  be  exhibited,  is 
made  of  maple,  in  two  pieces,  but  sliding  one  from  the  other, 
the  same  end  being  always  held  on  the  ground,  and  the  gradu- 
ations starting  from  that  point. 

The  graduations  are  made  to  tenths  and  hundredths  of  a  foot, 
the  tenth  figures  being  black,  and  the  feet  marked  with  a  large 
red  figure. 

The  front  surface,  on  which  the  target  moves,  reads  to  6£ 
feet ;  when  a  greater  height  is  required,  the  horizontal  line  of 
the  target  is  fixed  at  that  point,  and  the  upper  half  of  the  rod, 
carrying  the  target,  is  moved  out  of  the  lower,  the  reading 
being  now  obtained  by  a  vernier  on  the  graduated  side,  up  to 
an  elevation  of  12  feet. 

The  target  is  round,  made  of  thick  sheet  brass,  having,  to 
strengthen  it  still  more,  a  raised  rim,  which  also  protects  the 
paint  from  being  defaced. 

The  target  moves  easily  on  the  rod,  being  kept  in  any  posi- 
tion by  the  friction  of  the  two  flat  plates  of  brass  which  are 
pressed  against  two  alternate  sides,  by  small  spiral  springs, 
working  in  little  thimbles  attached  to  the  baud  which  surrounds 
the  rod. 

There  is  also  a  clamp-screw  on  the  back,  by  which  it  may  be 
securely  fastened  to  any  part  of  the  rod. 

The  face  of  the  target  is  divided  into  quadrants  by  horizontal 
and  vertical  diameters,  which  are  also  the  boundaries  of  the 
alternate  colors  with  which  it  is  painted. 


348  PLANE   SURVEYING. 

The  colors  usually  preferred  are  white  and  red ;  sometimes 
white  and  black. 

The  opening  in  the  face  of  the  target  is  a  little  more  than  a 
tenth  of  a  foot  long,  so  that  in  any  position  a  tenth  or  a  foot 
figure  can  be  seen  on  the  surface  of  the  rod. 

The  right  edge  of  the  opening  is  chamfered,  and  divided  into 
ten  equal  spaces,  corresponding  with  nine-hundredths  on  the 
rod  ;  the  divisions  start  from  the  horizontal  line  which  separates 
the  colors  of  the  face. 

The  vernier,  like  that  on  the  side  of  the  rod,  reads  to  thou- 
sandths of  a  foot. 

The  clamp,  which  is  screwed  fast  to  the  lower  end  of  the 
upper  slid  ing-piece,  has  a  movable  part  which  can  be  brought 
by  the  clamp-screw  firmly  against  the  front  surface  of  the  lower 
half  of  the  rod,  and  thus  the  two  parts  immovably  fastened  to 
each  other  without  marring  the  divided  face  of  the  rod. 

405.  The  Philadelphia  Rod.  This  rod  is  made  of  two 
strips  of  cherry,  each  about  f  inch  thick  by  l£  inches  wide  and 
7  feet  long,  connected  by  two  metal  sleeves,  the  lower  one  of 
which  has  a  clamping-screw  for  fastening  the  two  parts  together 
when  the  rod  is  raised  for  a  higher  reading  than  7  feet. 

Both  sides  of  the  back  strip  and  one  side  of  the  front  one 
are  planed  out  -fa  inch  below  the  edges ;  these  depressed  sur- 
faces are  painted  white,  divided  into  feet,  tenths  and  hundredths 
of  a  foot,  and  the  feet  and  tenths  figured. 

The  front  piece  reads  from  the  bottom  upward  to  7  feet, 
the  foot  figures  being  red  and  an  inch  long,  the  tenth  figures 
black  and  eight-tenths  of  an  inch  long.  When  the  rod  is 
extended  to  full  length,  the  front  surface  of  the  rear  half  reads 
from  7  to  13  feet,  and  the  whole  front  of  the  rod  is  figured 
continuously  and  becomes  a  self-reading  rod  13  feet  long. 

The  back  surface  of  the  rear  half  is  figured  from  7  to  13 
feet,  reading  from  the  top  down  ;  it  has  a  vernier  also  by 
which  the  rod  is  read  to  two-hundredths  of  a  foot  as  it  is 
extended.  The  target  is  round  and  made  of  sheet-brass,  raised 


FIELD   INSTRUMENTS.  349 

on  the  perimeter  to  increase  its  strength,  and  is  painted  in  white 
and  red  quadrants  ;  it  has  also  a  scale  on  its  chamfered  edge, 
reading  to  tvvo-hundredths  of  a  foot. 

When  a  level  of  less  than  7  feet  is  desired,  the  target  is 
moved  np  or  down  the  front  surface,  the  rod  being  closed 
together  and  clamped  ;  but  when  a  greater  height  is  required, 
the  target  is  fixed  at  7  feet  and  the  rear  half  slid  out,  the  scale 
on  the  back  giving  the  readings  like  those  of  the  target  to  two- 
hundredths  of  a  foot. 

This  rod  is  so  graduated  that  the  leveller  is  enabled  to  take 
the  reading  direct  from  it,  the  rodman's  duties  being  simply  to 
hold  the  rod  vertical  over  the  points.  It  is  hence  called  a  self- 
reading  or  speaking  rod. 

406.  The  Rod-Level.  The  figures  below  represent  a  level  re- 
cently devised,  for  the  more  accurate  plumbing  of  levelling-rods. 


ROD-LEVEL.  ROD-LEVEL  AS  APPLIED  TO  A  ROD. 

The  left-hand  figure  shows  it  when  folded  for  convenience  in 
carrying.  Its  convenience  and  value  commend  it  to  general 
favor. 

407.  Levelling  is  measuring  in  a  vertical  direction.  In  his 
treatise  on  levelling,  Frederick  W.  Simms  says:  "Levelling  is 
the  art  of  tracing  a  line  at  the  surface  of  the  earth  which  shall 


350  PLANE   SURVEYING. 

cut  the  directions  of  gravity  everywhere  at  right  angles.  .  .  . 
The  direction  of  gravity  invariably  tends  towards  the  centre 
of  the  earth,  and  may  be  considered  as  represented  by  a  plumb- 
line  when  hanging  freely,  and  suspended  beyond  the  sphere  of 
attraction  of  the  surrounding  objects.  .  .  .  The  operation  of 
levelling  may  be  defined  as  the  art  of  finding  how  much  higher 
or  lower  any  one  point  is  than  another,  or,  more  properly,  the 
difference  of  their  distances  from  the  centre  of  the  earth." 

A  surface  like  that  of  still  water  may  be  called  a  level  sur- 
face. The  curve  formed  by  the  intersection  with  such  a  sur- 
face of  a  vertical  plane  is  a  line  of  true  level;  a  line  tangent  to 
the  latter  is  a  line  of  apparent  level. 

Levelling  is  the  art  of  determining  the  differences  of  elevation 
of  two  or  more  points,  or  of  determining  how  much  one  point 
is  above  or  below  a  line  of  true  level  passing  through  the  other 
point. 

408.  From  the  foregoing  it  is  evident  that,  on  account  of  the 
curvature  of  the  earth,  a  horizontal  line  is  not  really  through- 
out its  length  a  level  line  ;  that  of  two  points  in  the  same  level 
line  each  will  have  its  own  horizon.     Hence,  in  levelling,  the 
effect  of  the  curvature  of  the  earth  upon  the  comparative  eleva- 
tions  of   different  points    must   be  taken    into   consideration. 
The  effect  of  the  curvature  is  to  make  objects  appear   lower 
than  they  really  are. 

The  air  nearer  the  surface  of  the  earth  is  denser  than  that 
farther  removed  from  the  surface.  This  difference  in  density, 
causing  refraction  of  light,  will  affect  the  elevation  of  a  point 
as  observed  through  the  telescope  of  a  level,  so  that  it  also 
must  be  taken  into  consideration.  Its  effect  is  to  make  objects 
appear  higher  than  they  really  are.  The  error  caused  by  refrac- 
tion is  one-seventh  as  great  as  that  caused  by  curvature. 

Let  us  first  find  an  expression  for  the  correction  due  to  the 
curvature  of  the  earth.  That  is  — 

409.  To  find  the  deviation  from  its  tangent  of  a  line  of  true 
level. 


LEVELLING. 


351 


Let  0  represent  the  centre  of  the  earth,  PN  a  line  of  true 
level,  and  PN'  its  tangent,  or  a  line  of  p  , 

apparent  level.  The  distance  NN'  cor- 
responding to  the  length  of  sight  PN  is 
required. 

From  Geometry, 

PN1'^  NN'(2  ON+NN')  ; 
PN~'2 


20N+NN' 

For  ordinary  distances,  the  length  of 
the  arc  may  be  regarded  as  that  of  the 
tangent,   and  NN'  as  inconsiderable  in 
comparison  with   2  ON,  the  diameter  of  u 
the  earth.     Therefore,  calling  the  length  of  sight  d,  the  cor- 
rection c,  and  the  radius  of  the  earth  r,  we  have 

d2 


and  the  correction  for  refraction 

~7C~7X  2r~14r' 

then  the  correction  due  to  curvature  and  refraction,  which  we 
will  call  (7,  is 

„_!„=£_.*., 

7        2r      Ur 


or, 


This  correction  must  be  added  to  the  height  of  the  object  as 
found  by  the  level. 

In  practice,  the  necessity  for  using  the  above  formula  is 
avoided  whenever  it  is  possible  to  set  the  level  at  equal  dis- 
tances from  the  points  whose  difference  of  height  is  required. 


352 


PLANE   SURVEYING. 


EXERCISES. 

1.  Assuming  the  diameter  of  the  earth  7,926  miles,  show  that 
for  a  mile  sight  c  =  about  8  inches.     Find  the  value  of  C  for  the 
same  distance. 

2.  What  is  the  correction  due  to  curvature  for  half  a  mile? 

3.  What  is  the  length  of  sight  when  C  equals  one-tenth  of  a 
foot? 

4.  Show  that,  practically,  the  correction  for  curvature  in  feet 
is  equal  to  two-thirds  the  square  of  the  distance  in  miles. 


410.  If  two  points  Jf,  JV,  whose  difference  of  elevation  is 
required,  can  be  observed  upon  from  some  point  P  about  equi- 
distant *  from  them,  not  necessarily  in  their  line,  set  up  the 
level  at  P,  and  note  the  reading  of  a  rod  held  vertically  over 
each  point.  The  difference  of  the  two  readings  will  indicate 
the  difference  of  level  required. 


411.  If  the  above  method  is  impracticable,  set  up  the  instru- 
ment at  some  point  P — either  in  or  out  of  the  line,  no  matter 
which  —  from  which  a  rod  may  be  observed  on  the  first  station 
M,  and  also  on  another  point  0  in  the  direction  of  N,  about  equi- 
distant with  M  from  the  instrument.  Remove  the  level  to  a 


*  Placing  the  instrument  in  this  position  lessens  the  effects  of  inaccurate 
adjustment  and  renders  unnecessary  the  corrections  indicated  in  Article 
409. 


LEVELLING.  353 

new  position  P',  whence  observe  again  the  rod  on  0,  also  the 
rod  reading  at  N. 

The  difference  between  the  readings  of  the  rod  at  M  and  0 
shows  how  much  higher  the  latter  is  than  the  former,  and  in  like 
manner  the  difference  of  the  readings  at  0  and  JV  gives  the  differ- 
ence in  elevation  of  these  points,  and  so  on,  no  matter  what  the 
number  of  stations.  The  difference  in  height  of  M  and  N 

=  Mm  —  Go  +  Oo'  —  Nn  ; 
or,  Mm  +  Oo'  —  Oo  -  Nn 

=  Mm  +  Oo'  —  (Oo  +  Nn) . 

Calling  Mm  and*  Go'  back-sights,  and  the  other  two,  fore- 
sights, we  perceive  that  the  difference  of  level  of  two  points  is 
shown  by  subtracting  the  sum  of  the  fore-sights  from  the  sum 
of  the  back-sights. 

412.  Again,  in  levelling,  we  measure,  by  means  of  the  rod, 
how  much  lower  than  the  line  of  sight  (height  of  instrument) 
certain  points  are.     Thus  we  may  determine  the  relative  eleva- 
tions of  the  points.     Suppose,  for  example,  it  be  required  to 
determine  the  difference  in  elevation  of  any  two  points.     For 
reasons  already  given,  set  the   level  equally  distant  from   the 
points.     If  this  cannot  be  done,  and  both  observations  have  to 
be  taken  from  one  of  the  stations,  especially  if  the  distance 
between  them  is  considerable,  correction  as  previously  described 
must  be  made.     But  in  this  case  suppose  it  is  possible ;  and 
suppose  that  when  held  on  one  point,  the  rod  reads  7.255  ;  that 
is,  this  point  may  be  considered  7.255  below  the  line  of  sight, 
and  4.755  when  held  on  the  other;  then  the  first  may  be  con- 
sidered 7.255  —  4.755,  or  2.500  farther  than  the  second  below 
the  line  of  sight,  or  lower  than  the  second. 

413.  Suppose  it  be  required  to  determine  the  difference  in 
elevation  between  two  points,  of  which  one  is  so  much  higher 
than  the  other  that  the  rod  is  too  short  to  give  a  reading  on 
both  points  for  one  position  of  the  instrument.     In  such  a  case 


354  PLANE   SURVEYING. 

one  or  more  auxiliary  points,  called  turning-points  (T.P.), 
must  be  used,  and  their  relative  elevations  determined.  Sup- 
pose the  reading  on  the  first  point  is  0.824,  and  on  a  turning- 
point  is  10.432  ;  the  latter  is  then  9.608  below  the  former.  Now 
the  instrument  must  be  moved  and  set  up  so  as  to  obtain  a 
reading  on  the  turning-point ;  and  (we  will  suppose)  on  the 
other  of  the  given  points.  Suppose  that  on  the  former  it  is 
1.302,  and  on  the  latter  8.634  ;  the  latter  is  then  7.332  below  the 
turning-point,  or  9.608  +  7.332,  or  16.940,  below  the  first  of  the 
two  given  points. 

The  first  sight  taken  after  setting  up  the  level  is  called  a 
back-sight,  or  plus  sight ;  those  taken  after  this,  and  before  the 
instrument  is  moved,  are  called  fore-sights  or  minus  sights.  As 
the  difference  of  the  readings  of  the  rod  on  two  points  gives 
their  difference  of  elevation,  the  difference  of  the  sum  of  the 
plus  sights,  and  the  sum  of  the  minus  sights  on  T.P.'s  and  the 
last  point  will  give  the  difference  in  elevation  of  the  extreme 
points.  In  the  above  example 

0.824  10.432 

1.302  8.634 


2.126  19.066 

19.066  -  2.126  =  16.940,  as  before. 

This  is  used  as  a  check  on  level-notes. 

In  extended  levelling,  permanent  elevations  fixed  during  the 
progress  of  the  work  for  future  reference  are  called  bench 
marks  or  benches  (B.M.). 

414.  In  levelling,  it  is  customary  to  refer  all  elevations  to  an 
assumed  level  plane,  called  the  plane  of  reference,  the  datum 
plane,  or  simply  the  datum.  Points  are  then  said  to  be  so 
much  above  or  below  the  datum.  As  this  plane  may  be  assumed 
at  pleasure,  it  is  generally  so  taken  as  to  be  lower  than  any 
point  whose  elevation  is  to  be  determined.  In  city  levelling 
this  plane  may  be  assumed  at  the  height  of  mean  low  water. 


LEVELLING.  355 

which  elevation  may  be  called  zero.  Then  a  point  which  has 
the  elevation  125.37  will  be  125.37  above  low  water. 

If  two  points  have  the  elevations  125.375  and  105.213  respec- 
tively, the  former  is  125.375  —  105.213,  or  20.162  higher  than 
the  latter. 

The  datum  having  once  been  determined,  its  elevation,  or 
that  of  a  point  a  known  distance  above  it,  should  be  perma- 
nently fixed  for  future  reference  and  comparison. 

415.  The  levels  for  profile  given  under  Street  Grades,  on 
page  365,  show  how  the  field  notes  in  levelling  ma}'  be  kept. 
The  elevation  of  the  bench-mark  from  which  they  start  is  51.415 
above  the  datum.  The  first  plus  sight  is  7.030,  which,  added  to 
51.415,  gives  58.445,  the  height  of  the  instrument  (H.I.)  above 
the  datum.  The  first  minus  sight,  which  is  on  a  turning-point 
(T.P.),  is  0.870,  which,  subtracted  from  58.445,  gives  57.575, 
the  height  of  the  T.P.  above  the  datum.  The  instrument  is 
then  moved,  set  up  again  in  a  convenient  place,  and  the  work 
proceeds. 

At  one  setting  of  the  instrument,  the  elevations  of  any  points, 
besides  the  turning-point,  which  are  not  too  high  or  too  low  to  be 
reached,  may  be  ascertained.  It  is  evident  that  if  any  error 
be  made  at  a  T.P.,  all  the  following  elevations  will  thereby  be 
affected ;  but  if  made  at  one  of  these  other  points,  only  the 
elevation  of  that  point  will  be  affected.  Hence  the  importance 
of  careful  observations  at  T.P's. 

In  the  above-mentioned  form  for  the  keeping  of  the  field 
notes,  all  the  observations  (Obs.)  are  set  in  one  column.  If 
desired,  plus  sights  and  minus  sights  may  be  set  in  different 
columns  ;  and  of  minus  sights,  those  on  turning-points  may  be 
set  in  a  column  by  themselves.  It  will  then  be  easy  to  apply 
the  check  before  described.  However,  the  form  given  is  in 
practice  very  convenient. 

EXERCISE. 

Tabulate  in  both  of  the  above  forms,  also  in  the  form 
headed 


356 


PLANE   SURVEYING. 


ELEVATION.      REMARKS 


the  following  level  notes  : 

Height  of  B.M 100.000. 

Obs.  on  B.M 5.132. 

"       "  Sta.   0 6.28. 

"       "     «*      1 7.12. 

"     '"     "      2 8.84. 

"  "  T.P.  3 9.780. 

From  new  position  of  inst.  obs.  on  Sta.  3,  2.160. 

Obs.  on  Sta.  4 5.89. 

"      "     "      5 7.92. 

"       "     "      6  .........  10.18. 

»  "  T.P.  7 12.020. 

Again  on  "  7 1.260. 

Obs.  on  Sta.  8 4.23. 

"      M  .>*...  9 5.87. 

"       "     "    10 6.94. 

416.  Wind  and  sunshine  affect  the  accuracy  of  levelling,  as 
of  work  with  the  transit.     For  very  good  work  it  is  desirable  to 
have  a  calm  day  on  which  the  sun  is  obscured  by  clouds.     In 
addition  to  a  proper  manipulation  of  the  instrument,  the  sights 
should  not  be  longer  than  from  200  to  300  feet,  the  rod  should 
be  held  vertical,  and  the  rod  man  should  select  for  turning-points 
good  and  firm  points  on  stones,  pegs,  etc.,  on  which  the  rod 
may  be  freely  turned  or  spun  around. 

417.  Numerous  bench-marks  should  be  located  in  convenient 
places.     In -a  city  such  places  are  at  the  intersections  of  streets, 
on  door-sills  of  buildings  which  have  become  thoroughly  settled, 
on  roots  of  trees,  etc.     There  are  many  other  suitable  places 
which  will  suggest  themselves. 


LEVELLING. 


357 


418.  In  city  work,  in  making  a  circuit  of  levels  for  the 
establishment  of  grade  elevations  and  bench-marks,  the  work 
should  check  out  with  no  greater  error  than  0.01  foot  in  three 
miles. 

In  levelling,  as  in  all  other  work,  regard  must  be  had  to  the 
difference  between  actual  mistakes,  the  results  of  carelessness, 
and  the  degree  of  accuracy  actually  obtainable  by  the  observer. 

We  will  now  describe  a  general  method  of  running  a  grade- 
line  with  the  transit.  In  the  figure  the  irregular  line  represents 
the  profile  of  the  ground,  and  the  straight  line  the  grade-line. 


Let  it  be  required  to  run  a  grade-line  from  ^4,  elevation  30.29, 
to  B,  elevation  28.79  ;  elevation  of  ping  or  ground  at  A  33.49, 
at  B  27.26  ;  therefore  cut  at  A  3.20  and  fill  at  B  1 .53. 

Set  the  transit  over  A ;  and,  using  the  long  level-tube,  take 
the  elevation  from  a  convenient  bench.  Suppose  the  H.I. 
is  found  to  be  38.21  ;  then  the  length  of  the  rod  for  marking 
the  grade-line  (called  working  height)  is  38.21—30.29  =  7.92. 
The  rod  will  then  be  taken  to  B  and  held  on  the  plug.  But  as 
the  plug  is  1.53  below  the  grade-line  at  B,  the  target,  when  the 
rod  is  held  for  grade  on  that  plug,  will  be  set  at  7.92  -f  1.53  = 
9.45.  When  thus  held,  the  observer  will  set  the  horizontal 
cross-hair  on  the  middle  of  the  target  and  clamp  the  telescope. 
The  line  of  sight  will  then  be  a  line  parallel  with  the  grade-line 
and  7.92  above  it.  Care  must  be  taken  to  use  the  rod  7.92,  and 


358  PLANE   SURVEYING, 

not  9.45,  as  the  working  height.  Measurements  may  now  be 
made  from  the  line  of  sight  to  determine  the  cut  to  the  grade- 
line  at  any  intermediate  point. 

Suppose  at  C  the  rod  read  5.97 ;  then  the  cut  at  that  point  is 
7.92-5.97  =  1.95. 

How  would  you  proceed  if  the  instrument  were  set  at  B1 
The  cuts  or  fills  to  grade  at  any  points  may  be  determined  by 
taking  the  elevations  of  the  ground  at  those  points  and  calcu- 
lating the  grade  elevations  at  the  same  points.     The  difference 
of  elevation  will  be  the  cut  or  fill  required. 

B.    OFFICE  INSTRUMENTS. 

419.  In  addition  to  the  various  drawing-instruments  previ- 
ously described  the  student  should  understand  the  use  of  that 
elegant  instrument  the  polar  planimeter.  In  ascertaining  the 
areas  of  figures  having  irregular  boundaries  it  will  be  found 
extremely  useful.  He  should  also  become  acquainted  with  the 
different  methods  for  the  rapid  reproduction  of  drawings. 


SECTION  II. 

WORK. 

420.  The  work  of  the  city  surveyor  may  be  divided  into  two 
classes :  first,  public  work,  or  that  which  he  is  called  upon  to 
perform   for  the   city   government ;    second,  private  work,  or 
that  which   he  performs  for  private  citizens.     The   former  is 
generally  connected  with  the  streets ;  the  latter,  with  the  prop- 
erty between  them. 

Again,  all  of  his  work  may  be  classed  as  field  work  or  office 
worlr,  the  former  of  which  we  will  now  consider. 

A.     FIELD    WORK. 

421.  Public  Work.      There   are    many  and   varied   natural 
features  and  artificial  influences  affecting  the  original  location 


FIELD    WORK.  359 

of  a  town  or  city.  To  the  thoughtful  student  many  of  these 
will  readily  suggest  themselves.  While  in  the  choice  of  a  site 
the  surveyor  may  have  a  voice,  it  is  more  than  probable  that 
his  work  will  commence  upon  a  site  already  selected.  We  will 
now  describe  some  of  his  more  important  duties  as  performed 
for  the  town  or  city  government. 

422.  Street  Lines.  The  city  consists  of  streets  for  public 
use,  and  of  the  blocks  bounded  by  them,  the  land  in  which  is 
divided  and  sold  to  individuals  for  their  private  use.  Hence 
we  have  first  to  consider  the  general  plan  or  arrangement  of 
the  streets,  their  widths  (the  distances  between  house  lines), 
and  their  distances  apart.  There  are  many  general  plans  which 
may  be  adopted,  or  may  be  used  as  the  foundation  for  new 
ones.  When  general  convenience  and  the  economical  division 
of  property  are  considered,  I  believe  there  is  none  which 
better  meets  the  requirements  than  that  which  is  characterized 
by  two  systems  of  parallel  streets  crossing  at  right  angles. 
With  this  general  arrangement,  and  some  well-located  diagonal 
avenues,  we  have  the  lay-out  of  a  beautiful  and  convenient 
city. 

The  general  directions  of  the  streets  should  be  such  that  the 
greatest  number  may  during  the  day  be  visited  by  the  sunshine. 
This  will  be  accomplished  if  one  set  of  parallel  streets  runs  in  a 
northeasterly  and  southwesterly  direction. 

Every  important  street  should  be  at  least  60  feet  wide,  while 
some  of  the  main  streets  should  be  at  least  100  feet  wide,  with 
avenues  even  wider.  The  streets  will  then  admit  freely  air  and 
sunshine,  which  latter  is  too  often  in  narrow  streets  cut  off  by 
tall  buildings  ;  while  the  avenues  will  be  in  harmony  with  their 
design  as  elegant  thoroughfares. 

Another  important  consideration  which  affects  the  width  of 
streets  is  the  expense  of  paving  and  of  keeping  them  in  order. 

The  distances  of  the  streets  from  each  other  will  vary  very 
much)  according  to  the  purposes  for  which  the  included  prop- 
erty is  to  be  used,  and  how  it  is  to  be  divided.  Thejf  may  vary 


360  PLANE   SURVEYING. 

from  300  to  600  feet.     The  sidewalks  will  be   from  one-fifth 
to  one-fourth  of  the  width  of  the  streets. 

In  small  towns  an  elaborate  design  will  not  be  attempted ; 
but  it  is  alwa3"s  best  to  have  in  view  the  possibilities  of  future 
growth. 

423.  With   the   transit',  the  surveyor  will  run  and  extend 
street  lines,  and  will  turn  off  required  horizontal  angles  on  the 
horizontal  graduated  circle  of  that  instrument.    It  is  convenient 
to  work  upon  the  centre  lines  of  the  streets.     Two  base  lines 
having  been  carefully  located  at  right  angles  with  each  other, 
the  centrelines  of  the  two  sets  of  streets  will,  with  the  most 
reliable  measuring-instruments  at  the  disposal  of  the  surveyor, 
be   carefully  located   parallel   with  them  respectively.     If   the 
land   is   quite   level,  a  200-foot   steel   tape  is   useful.      If   it 
be   inclined   and   irregular,   a    100-foot   tape   is   better   suited 
to  the  purpose.     In  any  case,  the  hand-level,  plummet,  ther- 
mometer, etc.,  should  be  used.     The  work,  like  all  work  of 
the  surveyor,   should  be  carefully   checked  by  a  test   of   the 
different  angles  and  distances.     All  this  work  should  be  done 
with   the   greatest   care.     It   is   desirable,  in   order   to   guard 
against  future  difficulties  in  regard  to  measurements  by  other 
pai-ties,    to   make   streets   and    block   distances   a  little    full; 
that  is,  greater   than    they  are  actually  required  to  be  —  say 
about  one-fourth  of   an  inch  in   100  feet.     As  the  work  pro- 
gresses, it   will   be   properly   marked   with   stakes,   as   before 
described.     After  the  satisfactory  location  of  the  centre  lines 
of  the  street,  the  house  lines  may  easily  be  located  therefrom. 

424.  The  work  of  the  surveyor  may  be  not  in  laying  out  and 
regulating  a  new  town,  but  in  connection  with  one  already  laid 
out.     The  extensions  of  the  old  town  may  be  carried  on  in 
harmony  with  the  plan  already  existing,  or  they  may  be  on  a 
plan  altogether  different,  and  after  the  manner  already  described 
for  a  new  town.     He  will  find  that  the  already  built-up  portions 
of  the  town  have  been  previously  regulated,  or  that  they  have 


FIELD    WORK.  361 

not  been.  If  they  have  been,  it  is  advisable  in  carrying  on 
the  work  therein  to  adhere  as  closely  as  possible  to  established 
lines,  elevations,  standard  of  measurement,  etc.,  lest  any  altera- 
tions should  lead  to  expensive  and  unnecessary  legal  complica- 
tions. If  the  town  has  never  been  regulated,  the  first  steps 
will  be  to  regulate  its  streets.  In  doing  this  a  complete  survey 
will  be  required.  Instrument  lines  will  be  carefully  located 
with  the  transit  on  all  streets,  and  the  angles  at  their  intersec- 
tions determined.  These  lines  will  be  the  basis  for  the  location, 
by  offsets,  of  all  buildings,  fences,  etc.  As  the  survey  goes  on, 
the  results  will  be  carefully  plotted  to  a  conveniently  large 
scale  ;  and  from  the  completed  plot,  an  advantageous  location  of 
the  streets  may  be  determined  upon.  They  will  then  be  located 
upon  the  ground  to  correspond.  All  important  measurements 
will  be  made,  as  before  described,  with  the  steel  tape,  with  all  the 
corrections  carefully  attended  to.  Offsets  to  fences,  etc.,  need 
not  be  made  with  so  much  care,  and  the  corrections  will,  as  a 
rule,  be  superfluous.  During  the  progress  of  the  work  in  an  old 
town,  as  in  a  new  one,  all  important  lines  will  be  carefully  marked 
with  stakes,  and  upon  permanent  objects,  as  houses,  etc. 

425.  The  streets  in  any  'city  or  town  having  been  satisfac- 
torily located  according  to  the  general  plan,  it  is  necessary,  in 
order  to  preserve  work  already  done,  and  to  prevent  conflict  in 
future  work,  that  the  location  of  the  street  lines  should  be  pre- 
served. On  account  of  the  perishable  nature  of  wooden  stakes, 
and  the  fact  that  they  may  soon  be  disturbed,  it  is  necessary 
to  use  something  more  permanent.  This  is  generally  found  in 
stones.  Mere  stones,  or  monuments  used  for  permanently  hold- 
ing the  lines  of  streets,  are  differently  located  and  are  of  differ- 
ent sizes,  depending  upon  then-  location.  Sometimes  they  are 
placed  in  the  sidewalks  5  feet  from  the  house  lines.  Then 
they  need  not  be  more  than  4  or  5  inches  square  and  2  feet 
in  length.  The  line  is  determined  by  a  small  hole  drilled  in  the 
top  of  the  stone.  Sometimes  the  top  of  the  stone  is  placed 
below  the  surface  of  the  pavement ;  sometimes  it  is  placed  flush 


362  PLANE   SURVEYING. 

therewith.  Larger  stones  set  in  the  intersections  of  the  streets, 
where  their  centre  lines  cross,  are  very  conveniently  situated 
for  use,  and  afford  a  very  satisfactory  means  of  marking  street 
lines.  On  account  of  their  more  exposed  position,  they  must 
be  larger  than  those  previously  described,  and  should  be  set 
with  the  greatest  care,  the  materials  around  them  being  well 
packed  and  rammed.  They  should  be  paved  about  and  well 
protected  from  danger  from  traffic.  The  stones  should  be 
square  in  cross-section  about  3  feet  long,  about  8  inches  square 
on  the  top,  and  about  1  foot  square  on  the  bottom,  the  top  and 
bottom  being  at  right  angles  with  the  axis  of  the  stone.  The 
line  is  determined  as  before  by  a  hole  drilled  in  the  top  of  the 
stone.  From  their  situation  we  call  these  stones  centre  stones. 
It  is  well  also  to  mark  substantial  buildings  standing  at  the 
corners  of  streets  with  their  distances  from  the  house  lines  of 
the  streets,  these  distances  having  been  carefully  determined 
by  measurements.  In  general,  a  line  having  once  been  deter- 
mined upon  as  satisfactory,  every  available  means  should  be 
employed  to  preserve  its  location,  as  any  change  would  ob- 
viously be  attended  with  inconvenience  and  danger. 

426.  Street  Grades.  In  the  selection  of  a  site  for  a  town, 
and  in  the  location  of  the  streets  of  a  town  or  city,  a  topo- 
graphical map  will  be  of  much  service.  This  map  will  show  at 
a  glance  the  shape  of  the  ground  under  consideration.  If  the 
surface  of  the  earth  were  cut  by  horizontal  planes  5,  10,  20,  or 
more  feet  apart,  and  the  curves  in  which  these  planes  intersect 
the  surface  were  projected  upon  a  horizoual  plane,  the  resulting 
lines  would  be  called  contour  lines  or  contours.  These  curves 
would  represent  points  of  the  same  elevation.  Their  distances 
apart  would  represent  relative  inclination  in  the  ground,  the 
curves  being  nearer  as  the  ground  is  steeper.  The  determina- 
tion of  these  contours  is  an  important  feature  in  topographical 
surveying.  In  addition  to  its  other  uses,  such  a  map  would  be 
of  service  in  locating  sewers,  also  in  fixing  proper  elevations 
and  grades  for  streets.  The  field  work  necessary  in  the  prep- 


FIELD    WORK.  363 

aratiou  of  topographical  maps,  which  we  will  briefly  notice,  may 
bo  done  as  follows  :  Two  sets  of  parallel  lines  having  been 
located  at  right  angles  with  each  other  by  means  of  the  transit 
and  tape,  the  level  will  be  set  up,  and  a  number  of  points  at 
any  one  elevation  above  the  datum  found  with  the  level  and 
the  rod,  and  their  locations  with  reference  to  the  two  sets  of 
lines  determined.  Another  set  of  points  as  far  above  or  below 
the  former  as  the  planes  are  apart  will  in  like  manner  be  deter- 
mined and  located,  and  so  on  until  the  entire  ground  has  been 
gone  over.  The  above  method  of  topographical  surveying  in 
determining  contours  is  not  a  very  rapid  one.  The  stadia 
method  is  more  rapid,  and  is  well  adapted  to  large  areas.  In 
addition  to  the  usual  horizontal  cross-hair  in  the  transit,  two 
others  are  introduced,  one  above  and  one  below  the  former. 
The  instrument  has  also  a  vertical  circle.  The  stadia-hairs  are 
so  arranged  that  when  the  level  rod  is  held  at  a  certain  distance 
from  the  transit,  a  certain  number  of  feet  on  the  rod  is  included 
between  them.  The  distance  of  any  point  from  the  instrument 
can  be  determined,  as  it  varies  with  the  number  of  feet  inter- 
cepted on  the  rod.  The  line  of  sight  must  be  at  right  angles 
to  the  rod ;  if  it  is  not,  a  calculation  must  be  made  to  deter- 
mine the  distance.  By  this  distance  and  a  horizontal  angle  the 
point  is  located  horizontally.*  The  elevation  of  the  point  above 
the  station  at  which  the  instrument  is  placed  is  obtained  by 
observing  on  the  rod  a  point  as  much  above  the  ground  as  the 
telescope  is,  and  taking  the  vertical  angle.  The  product  of  the 
horizontal  distance  and  the  tangent  of  the  angle  will  give  the 
required  difference  in  elevation.  The  plane  table  also  has  been 
much  used  in  making  topographical  surveys. 

Street  grades  themselves  will  be  determined  upon  in  the 
office,  after  the  necessary  data  has  been  obtained  in  the  field 

427.  A  very  convenient  method  of  obtaining  the  data  neces- 
sary for  the  determination  of  elevations  and  grades  for  the 
streets  is  to  obtain  a  continuous  profile  of  the  ground  on  the 


*  See  Chapter  II.,  Stadia  Measurements,  Articles  148-152. 


304  PLANE   SURVEYING. 

centre  line  of  each  street.  The  work  is  done  in  the  following 
manner :  The  level  having  been  set  up,  and  the  height  of 
instrument  determined  from  a  convenient  bench-mark,  an 
elevation  will  be  taken  on  a  level  plug  set  at  the  intersection 
of  the  centre  lines  of  two  streets.  Elevations  will  then  be 
taken  at  stations,  say  50  feet  apart,  about  on  the  centre  line, 
'  measurements  with  the  tape  being  commenced  at  the  inter- 
section before  mentioned,  and  made  carefully  enough  to  avoid 
any  error  that  might  affect  the  work.  In  addition  to  the  eleva- 
tions at  the  stations,  elevations  should  be  taken  at  any  interme- 
diate points  where  the  shape  of  the  ground  abruptly  changes ; 
and  the  points  should  be  located  by  measurement.  These 
intermediate  points  are  called  pluses.  When  the  next  inter- 
section is  reached,  measurements  will  be  commenced  anew,  and 
the  levelling  continued  in  the  same  manner.  Elevations  on 
level  plugs  at  intersections,  on  turning-points,  and  on  benches, 
which,  if  not  previously  established  should  be  established  as  the 
work  progresses,  should  be  carefully  taken  with  the  target. 
The  elevations  for  the  profile  should  be  read  without  the  target 
to  the  nearest  hundredth.  Such  circuits  should  be  made  in 
levelling  for  profiles,  and  the  levelling  on  the  cross-streets 
should  be  so  carried  on  as  to  check  the  work  in  every  wa}7. 
The  level  notes,  taken  as  described  for  the  profile  of  the  centre 
line  of  a  street,  are  shown  below.  They  are  from  actual  prac- 
tice. The  datum  is  mean  low  water  in  the River,  the 

elevation  of  which  is  taken  as  zero.  The  manner  of  plotting 
these  notes,  and  of  determining  grade  lines  is  given  under  the 
head  Office  Work. 

428.  In  order  to  avoid  errors  in  giving  grade  lines,  the 
grade  elevations  at  the  intersections  of  streets  should  be  per- 
manently marked.  This  may  be  done  by  placing  the  centre 
stones  before  described  so  that  their  tops  shall  be  at  the  grade 
elevation.  In  order  to  preserve  these  elevations  in  case  of  the 
removal  or  disturbance  of  the  stones,  bench-marks  should  be 
established  on  convenient  door-sills,  and  in  other  safe  and  con- 


FIELD    WOltK. 


365 


LEVELS   ON   FIFTH   AVENUE,    SOUTHERLY  FROM  MARY- 
LAND  AVENUE. 


FOR  PROFILE. 


Nov.  21,  1880,  A.M. 


STA. 

OBS. 

H.I. 

EL. 

REMARKS. 

B.M. 

51.415 

On  west  end  of  door-sill,  etc. 

+ 

7.030 

^58.445 



sp- 

0.870 

57.575 

1+ 

10.005 

67.580 

... 

B.M.  &(P. 

1.300 

66.280 

(  On  highest  point  of    red 
|      rock,  etc. 

i+ 

0.900 

67.180 

Sta.  0. 

0.000 

67.180 

(  Plug   middle   of  5th   and 
\      Md.  Aves. 

0  +  25. 

1.55 



65.63 

0  +  35. 

0.28 

66.90 

1. 

1.50 



65.68 



(  50-ft.    Sta.    meas.    south 
|      from  mid.  of  Md.  Ave. 

2. 

3.91 

63.27 

3. 

0.20 



60.98 

4. 

8.83 

58.35 

5. 

11.80 



55.38 

6. 

13.20 

53.98 

7. 



Plug  &  (  P. 

11.352 

55.828 

.... 

(  Plug  centre  5th  Ave.  and 
|      Anchorage  St. 

i  + 

4.365 

60.193 

B.M. 

5.480 

54.713 



j  Temporary  —  on  plug  near 
I      fence,  etc. 

Sta.  1. 

5.13 

55.06 

(  50-ft.  sta.  meas.  south  from 
(      middle  of  Anchorage  St. 

2. 

4.65 

55.54 

3. 

4.93 

55.26 

4. 

5.69 

54.50 

6. 

7.26 

52.93 

6. 
Plug  0  +  34. 

11.00 
12.224 

.... 

49.19 
47.969 

(  Plug  centre  6th  Ave.  and 
I      Brown  St. 

366  PLANE   SURVEYING. 

venient  places.  Besides  serving  as  benches  for  the  stones, 
these  bench-marks  will  be  used  in  doing  very  close  final  level- 
ling, the  tops  of  the  stones  being  too  uneven  for  that  purpose. 

429.  Marking  of  Lines  and  Grades.     The  lines  and  grades 
of  the  streets  having  been  finally  determined,  and  the  means  of 
preserving  them  having  been  established,  the  marking  of  these 
lines  and  grades  for  any  public  work,  as  street  extension  and 
grading,  curb  setting,  sewer  and  water-pipe  laying,  etc.,  can 
be   readily  done.     Street  lines  will  be  run  with   the  transit ; 
and,  in  the  manner  previously  described,  grade  lines  will  be  run 
with  the  same  instrument.     The  marking   of   street  lines  and 
grades  for  the  purposes  mentioned,  the  giving  of  lines  and  ele- 
vations for  other  public   work,  and  measurements  of  various 
kinds,   as  of  earthwork,   constitute  the   principal  part  of  the 
field  work  to  be  done  for  the  town  or  city  government  by  the 
city  or  town  surveyor ;  or,  as  the  officer  who  does  this  work 
may   have   more   extended  duties,   the   principal  part   of  the 
surveying  to  be  done  by  the  city  engineer. 

430.  Private  Work.     Continuing  the  description  of  the  field 
work  of  the  town  or  city  surveyor,  we  will  notice  the  second 
general  class  in  which  his  work  is  comprised ;  that  is,  work  for 
individuals,  or  private  work.     In   general,  —  for  other  duties 
in  this  connection  will  fall  to  his  lot,  such  as  surveying  large 
tracts   according   to   methods    already   described,    etc.,  —  this 
work  will   consist   in    marking   property   lines   and   in   giving 
grades  and  elevations.     As  a  rule,   in    a   town   or  city  more 
property  lines  are   marked  for  buildings  than  for  any  other 
purpose.     When   the   survevor   is   called   upon    to   locate    the 
lines  of  a  lot,  his  first  inquiry  will  be  as  to  the  data  by  which 
to  locate  them.     It  is  of  course  understood  that  in  this  con- 
nection the  only  power  of  the  surveyor  is  to  locate  lines  accord- 
ing to  given  data,   not,  as  man}'  persons  seem    to   think,  to 
establish  of  his  own  volition  new  lines.     So  we  will    inquire 
what  is  proper  data  for  locating  such  lines.     In  general,  the 


J.I 


24'6"  r    24V 


v  r 


FIKLD   WORK.  369 

party  desiring  to  have  the  lines  of  a  lot  marked  will  produce 
his  deed  for  the  property.  The  young  surveyor  will  be  inclined 
to  think  that  the  distances  given  in  deeds  are,  as  to  the  loca- 
tion of  lines,  final.  This  is  not  always  the  case.  When  walls, 
alleys,  stones,  and  other  permanent  landmarks  are  called  for, 
and  can  be  found,  they  will  take  precedence  of  distances  in 
locating  lines.  When  walls,  fences,  and  other  holdings  prove 
undisputed  possession  for  a  period  of  years,  though  they  may 
not  be  described  in  the  deed,  they  govern.  In  such  cases  it 
would  be  superfluous  to  mark  lines.  In  towns  and  cities  lots 
are  now  as  a  rule  located  from  the  streets.  Let  us  take,  in 
marking  the  lines  for  a  lot,  an  example  from  actual  practice. 
The  description  taken  from  the  deed  is  definite,  and  is  as 
follows : 

Beginning  at  the  easterly  side  of  West  Street,  between 
Eighth  and  Ninth  Streets,  at  the  distance  of  223  feet  from  the 
southerly  side  of  Ninth  Street;  thence  easterly,  parallel  with 
Ninth  Street,  132  feet  to  a  corner;  thence  southerly,  parallel 
with  West  Street,  28  feet  to  a  corner ;  thence  westerly,  par- 
allel with  the  first-described  line  and  Ninth  Street,  132  feet  to 
the  aforesaid  easterly  side  of  West  Street ;  and  thence  thereby, 
northerly,  28  feet  to  the  place  of  beginning.  The  lot  is 
located  as  shown  in  the  sketch.  The  owner  desired  to  have 
marked  upon  the  ground,  for  use  in  building,  the  two  lines  par- 
allel with  Ninth  Street  and  the  line  of  the  easterly  side  of 
West  Street.  In  order  that  they  may  not  be  removed  in 
making  excavations  for  cellars,  walls,  etc.,  the  nail  plugs  to 
mark  the  lines  are  set  3  or  4  feet  outside  of  the  lot.  In  the 
sketch,  S,  S,  S,  S  represent  the  stone  monuments  set  at  the 
intersections  of  the  centre  lines  of  the  streets  to  mark  lines 
and  grade  elevations.  Each  street  is  49  feet  wide.  In  mark- 
ing the  lot,  points  p,  p,  will  be  taken  in  the  centre  line  of  Ninth 
Street.  From  these  points  (if  there  are  no  obstructions  that 
prevent)  measurements  will  be  made  parallel  with  West  Street. 
Twenty-four  feet  six  inches,  half  the  width  of  Ninth  Street, 
and  223  feet,  the  distance  from  the  southerly  side  of  Ninth 


370  PLANE   SURVEYING. 

Street  to  the  northerly  side  of  the  lot,  will  be  laid  down,  and 
nails  placed  in  nail  plugs  at  a,  a,  to  mark  the  northerly  line 
of  the  lot.  From  these  the  southerly  line  will  be  located.  In 
a  similar  manner  the  front  and  back  lines  will  be  located. 
Lines  strained  from  a  to  a  and  from  &  to  6  will  cross  at  c, 
giving  a  corner  of  the  lot,  the  nail  plugs  being  undisturbed  as 
the  work  of  building  progresses. 

If,  on  account  of  impassable  obstacles,  as  buildings,  walls, 
etc.,  a  measurement  cannot  be  made  from  Ninth  Street  to  the 
place  for  the  nail  plug  a  back  o£  the  lot,  the  marking  Of  the  side 
lines  will  be  done  as  follows :  The  southeast  angle  at  the  inter- 
section of  Ninth  and  West  Streets,  89°  51',  if  not  known,  will 
be  taken.  In  addition  to  the  points  taken  in  the  centre  line 
of  West  Street  for  use  in  locating  the  front  and  back  lines  of 
the  lot,  an  additional  point  p'  will  be  taken,  and  at  this  point 
the  angle  89°  51'  will  be  thrown  in,  and  the  random  line  p'p11 
located  parallel  with  Ninth  Street.  On  this  random  line  points 
for  the  location  of  the  side  lines  will  be  taken.  Now,  suppose 
the  point  p'  is  found  by  measurement  to  be  257  feet  and 
6  inches  from  the  centre  of  Ninth  Street  (all  corrections  having 
been  made) ,  or  233  feet  from  the  southerly  side  thereof.  Then 
the  northerly  side  line  will  be  located  by  measuring  northerly 
from  the  line  p'p"  10  feet,  and  the  southerly  side  line  by 
measuring  southerly  from  the  line  p'p"  18  feet.  If  the  sur- 
veyor is  in  possession  of  an  instrument  thoroughly  reliable  for 
use  in  angular  measurements,  the  latter  method  of  marking 
side  lines  is  to  be  preferred.  When  one  measurement  is  made 
along  a  sidewalk  where  there  are  no  obstructions,  and  the 
other  through  fences  and  over  various  obstructions,  it  is  hardly 
possible  to  obtain  the  degree  of  accuracy  that  may  be  obtained 
by  the  angular  method.  Sometimes  it  may  be  necessary  to 
turn  off  an  angle  from  the  random  line  in  order  to  locate  the 
back  line  of  a  lot.  The  location  of  lines  is  often  marked  by 
nails  in  fences,  measurements  to  houses,  walls,  etc.,  instead 
of  by  nails  in  plugs. 

After  the  street  lines  have  been  located   and  marked,  the 


OFFICE   WORK.  371 

work  in  each  block  should  be  done  independently  of  the  other 
blocks. 

In  the  intervals  between  routine  work  it  is  desirable,  in  connec- 
tion with  gathering  other  data,  to  take  and  record  in  a  suitable 
.  book,  for  use  as  described  above,  the  angles  at  the  intersections 
of  the  streets,  thus  saving  time  in  marking  the  lines  of  lots. 

The  location  from  the  deed  of  the  lines  of  a  lot  is  not  always 
so  easy  as  in  the  example  given.  It  is  frequently  the  case 
that  the  distances  given  are  indefinite ;  sometimes  none  are 
given.  In  such  eases,  in  the  absence  of  established  holdings, 
or  other  means  of  determining  the  location  of  property  lines, 
the  matter  must  be  settled  by  an  arrangement  between  adjoin- 
ing owners. 

In  some  cases  a  lot  is  described  in  whole  or  part  without 
distances,  but  as  bounded  by  the  property  of  other  owners. 
In  such  a  case  the  location  of  the  lines  may,  if  the  descriptions 
in  the  deeds  of  these  other  proprietors  are  sufficiently  definite, 
be  determined  by  marking  the  lines  of  the  other  lots. 

431.  The  city  or  town  surveyor  will   frequently  be  called 
upon  for  surveys  to  locate  new  lines  with   reference  to  the 
street  lines,  or  for  surveys  of  tracts  of  land  in  or  adjoining 
the  city  or  town.     In  such  cases  his  manner  of  working  will 
be  based  upon  the  methods  of  land-surveying  already  described. 

Private  parties  will  frequently  require,  for  use  in  building 
operations,  the  marking  of  grade  lines.  This  will  be  done  in 
the  manner  previously  described.  In  marking  the  grade  and 
height  of  the  building  line  in  front  of  a  lot,  it  will  very  often 
be  found  convenient  to  mark  the  tops  of  the  front  line  plugs 
as  so  much  above  or  below  grade  elevation. 

B.     OFFICE   WORK. 

432.  Like  the  field  work,  the  office  work  of  the  surveyor  may 
be  classified  as  Public  Work  and  Private  Work. 

433.  Public  Work.     All   field  notes   should  be  sufficiently 
elaborate  to  be  understood  by  those  who  may  have  occasion  to 


372  PLANE   SURVEYING. 

refer  to  them.  They  should  be  carefully  arranged  and  indexed 
like  all  other  office  records  for  convenient  reference.  Plots  of 
work  should  be  made  whenever  they  will  aid  in  the  preservation 
and  proper  understanding  of  work  done  in  the  field.  When 
plans  are  sent  from  the  office,  copies  should  always  be  retained. 

434.  It  is  desirable  that,  besides  the  necessary  general  plans 
of  the  town  or  city,  the  surveyor  should  have  in  his  office  two 
sets  of  plans,  of  a  size  convenient  for  handling,  representing 
the  city  in  sections.  For  these  plans  a  horizontal  scale  of  100 
feet  to  the  inch  is  suitable. 

The  first  set  should  represent  street  lines.  On  them  should 
be  placed  all  tha  street  lines,  and,  in  figures,  the  widths  of 
streets  and  block  distances,  also  the  location  of  street  monu- 
ments, measurements  made  from  time  to  time  between  centres, 
angles  at  the  intersections  of  the  centre  lines  of  streets,  and 
any  other  data  of  a  like  nature  giving  information  in  regard  to 
horizontal  measurements,  whether  of  lines  or  angles. 

The  second  set  should  represent  street  grades.  On  them 
should  be  placed,  as  on  those  of  the  other  set,  the  street  lines 
and,  in  figures,  the  widths  of  streets,  block  distances,  and 
location  of  street  monuments.  In  addition,  there  should  be 
placed  upon  them  the  profiles  of  the  centre  lines  of  the  streets. 
These  plans  will  be  used  in  determining  grade  lines  for  the 
streets,  which,  after  they  have  been  determined,  will  be  placed 
upon  the  plans,  with  the  grade  elevations  (G.E.)  and  surface 
elevations  at  the  intersections  of  the  centre  lines  of  streets, 
grade  elevations  at  curb  corners,  and  any  other  data  giving 
information  in  regard  to  vertical  measurements.  The  street 
lines  having  been  laid  down,  we  will  explain,  in  connection 
with  the  accompanying  sketch  copied  from  a  plan  in  actual  use, 
how  the  data  given  on  page  365  would  be  used  in  placing  upon 
the  plan  the  profile  of  the  centre  line  of  Fifth  Avenue,  and 
then  how  the  plan  would  be  used  in  determining  suitable  grades 
for  the  streets. 


OFFICE   WORK.  373 

435.  If  the  points  whose  elevations  have  been  determined  by 
the  level  be  connected  by  a  line  in  a  vertical  plane,  such  a  line 
is  called  a  profile.     The  block  distance  from  Maryland  Avenue 
to  Anchorage  Street  is  297  feet  and  9  inches,  from  Anchorage 
Street  to  Brown  Street  is  294  feet,  from  Cedar  Street  to  Fifth 
Avenue  is  264  feet,  and  from  Fifth  Avenue  to  Sixth  Avenue  is 
160  feet.      Maryland  Avenue  is   64   feet  and  6   inches  wide, 
Anchorage  and  Brown  Streets  each  40  feet  wide,  and   Cedar 
Street,  Fifth  Avenue,  and  Sixth  Avenue  each  50  feet  wide. 
The  sidewalks  on 'Cedar  Street  and  on  Fifth,  Sixth,  and  Mary- 
land Avenues  are  12  feet  and  9  inches  wide,  and  on  Anchorage 
and  Brown  Streets  are  10  feet  wide.     By  the  use  of  the  profile 
of  Fifth  Avenue  we  will  illustrate  how  the  profiles  of  the  centre 
lines  of  the  streets  are  placed  upon  the  plan.     The  irregular 
lines  represent  profiles.     The  profile  is  commenced  by  consider- 
ing the  centre  line  of  Fifth  Avenue,  as  drawn  on  the  plan,  to 
have  the  elevation  67.180,  which  is  the  elevation  in  the  notes 
for  the  surface  of  the  ground  at  the  intersection  of  the  centre 
lines  of  Fifth  and  Maryland  Avenues.     The  stations  and  pluses 
as  given  in  the  notes  are  then  laid  down  by  scale  on  the  centre 
line  of  Fifth  Avenue,  in  the  order  in  which  they  were  taken  in 
the  field,  beginning  at  the  centre  of  Maryland  Avenue.     The 
elevation  at  each  of  the  points  thus  located  is  then  plotted,  in  a 
perpendicular  to  the  centre  line  at  that  point,  with  reference 
to  the  centre  line  elevation  67.180.     In  this  case  the  points  ob- 
tained will  all  fall  below  the  centre  line.     These  points  are 
points  in  the  profile,  and,  being  joined,  will  give  the  profile  as 
shown.     The  profile  of  Fifth  Avenue  having  been  started  at 
the  elevation  of  the  ground  at  the  intersection  of  Fifth  and 
Maryland  Avenues,  is  said  to  be  swung  on  Maryland  Avenue. 
In  the  sketch,  the  profiles  of  Cedar  Street  and  Sixth  Avenue 
also  are  swung  on  Maryland  Avenue.    Those  of  Anchorage  and 
Brown  Streets  are  swung  on  Cedar  Street. 

436.  A  little  thought  will  make  it  evident  to  the  student  that, 
as  the  differences  of  elevation  are  small  as  compared  with  the 


374  PLANE   SURVEYING. 

horizontal  distances,  if  both  were  plotted  to  the  same  scale,  or, 
as  we  say,  if  the  vertical  and  horizontal  scales  were  made  equal, 
the  differences  in  elevation  will  scarcely  be  apparent.  This  is 
remedied  bv  conveniently  exaggerating  the  vertical  scale.  For 
example,  if  the  horizontal  scale  be  made  100  feet  to  the  inch, 
the  vertical  scale  might  be  made  10  feet  to  the  inch.  In  the 
sketch  the  two  scales  have  this  ratio. 

EXERCISE.  Let  the  student  select  scales,  and,  in  the  manner 
described  above,  prepare  a  profile  from  the  field  notes  given  on 
page  356. 

437.  Having  thus  plotted  the  streets  and  profiles  in  a  large 
area,  we  may,  by  use  of  the  plan  thus  made,  determine  suitable 
grades  for  the  streets.  This  will  involve  careful  study  of  the 
shape  of  the  ground,  location  of  watercourses,  probable  loca- 
tion of  sewers,  and  effect  upon  property.  The  effect  of  a  pro- 
posed grade  for  one  street  upon  those  which  it  crosses  must  be 
particularly  noticed.  To  properly  perform  this  work  involves 
that  knowledge  and  judgment  which  can  only  be  acquired  by 
long  experience.  The  straight  lines  drawn  in  connection  with 
the  profiles  represent  the  surface  grades  of  the  finished  streets. 
In  fixing  the  grade  for  Fifth  Avenue,  those  of  the  other  streets 
having  been  taken  into  consideration,  it  was  found  best  to  have 
a  cut  of  2  feet  at  Maryland  Avenue,  no  cut  or  fill  at  Anchorage 
Street,  and  a  cut  of  2  feet  at  Brown  Street.  The  elevations 
of  the  surface  at  the  intersections  of  Fifth  Avenue  with  Mary- 
land Avenue,  Anchorage  Street  and  Brown  Street,  are  respec- 
tively 67.180  on  the  ground,  55.828  and  47.969  on  plugs  flush 
with  the  ground.  The  grade  line  having  been  fixed,  the  grade 
elevations  (G.E.)  at  the  centres  are  respectively  65.180,  55.828, 
and  45.969,  and  the  descents  9.35  feet  and  9.86  feet,  as  shown 
in  the  sketch.  The  nature  of  grades  will  depend  much  upon 
local  considerations.  Grades  should  always  be  steep  enough  to 
secure  proper  drainage.  The  inclination  should  not  be  less 
than  1  in  100.  Considering  the  accumulations  of  dirt  on  many 
of  our  city  streets,  from  1  to  1.5  in  100  is  to  be  preferred. 


OFFICE   WORK.  377 

438.  In  streets  in  which  surface  water  is   carried  on   the 
streets,  some  streets  will  carry  the  water  in  gutters  across  oth- 
ers.    In  the  sketch  such  streets  are  indicated  by  having  arrows 
drawn  in  their  directions  across  intersections.     In  this  manner 
Fifth  Avenue  carries  the  water  across  Brown  Street,  and  An- 
chorage Street  carries  it  across  Fifth  Avenue.     The  water  flow- 
ing on  Fifth  Avenue,  from  Maryland  Avenue  towards  Anchorage 
Street,  will  turn  into  Anchorage  Street.     The  opposite  side  of 
Anchorage  Street,  at  the  house  line,  will  be  a  knuckle  as  high  as 
the  centre  of  the  street ;  and  the  water  will  flow  from  that  point 

i  towards  Brown  Street.  In  fixing  grades  great  care  must  be 
taken  to  so  arrange  them  that  one  street  shall  not  be  overtaxed 
with  water  from  the  others.  An  outlet  for  the  surface  water  is 
formed  in  the  natural  watercourses. 

If  the  grade  of  Anchorage  Street  were  very  heav}',  so  that  if 
continued  across  it  would  make  one  side  of  Fifth  Avenue  much 
higher  than  the  other,  it  would  be  desirable  to  break  the  grade 
of  Anchorage  Street  at  the  curb  lines  of  Fifth  Avenue,  giving 
only  sufficient  fall  to  carry  the  water  across  the  Avenue. 

439.  If  the  section  is  sewered,  and  if  the  sewers  are  made 
"large  enough  to  carry  the  surface  water,  the  gutters  across  the 
streets  will  be  dispensed  with,  and  inlets  to  the  sewers  placed 
at  the  curb  corners  of  the  blocks. 

440.  It  is  often  convenient  and  useful  to  have  plotted  on 
separate  streets  the  profile  and  grades  of  each  street. 

441.  Besides  making  street  and  grade  plans,  it  will  be  a  part 
of  the  office  work  of  the  surveyor  to  plot,  in  the  usual  manner 
of  plotting  such  work,  the  surveys  made  in  and  about  the  city 
or  town,  for  both  the  city  and  individuals. 

442.  In  some  cities  a  registry  of  property  is  kept.     The 
plotting  of  lots  in  suitable  record  books,  and  the  keeping  up  of 
the  records,  will  be  a  part  of  the  city  surveyor's  work. 


378  PLANE   SURVEYING. 

443.  Private  Work.     This  includes  the  preparation  of  any 
plans  ordered  for  their  own  use  by  parties  other  than  those  con- 
nected with  the  city  government. 

CONCLUSION. 

444.  The  student  must  bear  in  mind  that  he  can  never,  from 
books,  learn  to  be  an  accomplished  surveyor.     The  practice  is 
ever  in  advance  of  the  books.     Though  he  should  store  his 
mind  with  book  knowledge  upon  the  subject,  he  will  yet  be 
wanting  in  the  knowledge  and  readiness  regarding  actual  work 
which  can  only  be  acquired  by  a  long  experience.     Many  oper- 
ations which  can  with  difficulty  be  understood  from  pages  of 
explanation,  will,  when  their  actual  performance  is  seen,  be 
comprehended  in  a  short  time.     Again,  there  is  that  which  can 
never  be  learned  from  books  ;  that  is,  the  judgment  which  must 
be  constantly  exercised  in  practising  the  delicate  duties  of  a  city 
surveyor.     Among  other  things,  this  judgment  will  teach  him 
to  be  very  cautious  about  giving  voluntary  advice,  and  careful  in 
giving  even  that  which  is  requested ;  to  perform  his  duties  con- 
scientiously, and  to  keep  clear  of  all  entangling  alliances.     Let 
him  learn  everything  connected   with  a  complete  performance 
of  his  work,  from  the  work  of  the  axeman  up ;  that,  when  he 
directs,  he  may  do  it  with  the  same  grace  with  which  he  should 
ever  follow  the  directions  of  his  superiors. 

The  practice  of  city  surveying  is  a  most  excellent  drill.  If 
conscientiously  performed,  it  will  develop  careful  and  thought- 
ful habits.  However,  in  practice  the  student  will  also  have  to 
learn  to  avoid  "  fussing"  over  work,  and  to  proportion  to  the 
importance  of  the  work  in  hand  the  time  and  care  spent  upon  a 
particular  work. 

BOOKS. 

445.  Valuable  information  regarding  the  matters  treated  of 
in  this  chapter  will  be  found  in  the  following  publications : 

The  manuals  and  catalogues  of  instrument-makers. 


BOOKS.  379 

"A  Treatise  on  the  Principles  and  Practice  of  Levelling,"  by 
Frederick  W.  Simms ;  published  by  D.  Van  Nostraud,  New 
York. 

"A  Descriptive  Treatise  on  Mathematical  Drawing-Instru- 
ments," by  William  F.  Stanley  ;  published  by  E.  &  F.  N.  Spon, 
New  York  and  London. 

"  A  Manual  of  Drafting  Instruments  and  Operations,"  by  S. 
Edward  Warren  ;  published  by  John  Wiley  &  Son,  New  York. 

"  The  Draughtsman's  Handbook  of  Plan  and  Map  Drawing," 
by  George  S.  Andre" ;  published  by  E.  &  F.  N.  'Spon,  New 
York  and  London. 

The  student  of  surveying  who  wishes  to  extend  his  studies 
into  the  field  of  city  engineering  will  find  information  upon 
that  subject  in  the  numerous  works  upon  its  special  branches, 
and  in  the  current  technical  periodicals  of  that  class.  Much 
information  regarding  present  American  practice  in  city  engi- 
neering will  be  found  in  the  series  of  papers  on  "  Municipal 
Engineering"  now  being  published  in  '-Engineering  News." 
When  completed,  these  in  book  form  will  make  a  very  useful 
v  olume. 


CHAPTER  VIII. 

MINE  SUKVEYING. 

446.  The  survey  of   underground   excavations    (mines)    to 
determine  their  position  and  extent  may  be  principally  for  the 
purpose  of  projecting  the  points  upon  a  horizontal  plane  as  in 
land  surveying. 

But  in  strata  of  high  inclination  and  in  cavernous  spaces 
various  vertical  projections  will  be  needed  to  complete  the 
graphical  representation  of  the  workings  ;  and  in  fissure  veins 
the  elevation  may  be  more  important  than  the  plan. 

447.  Surveys  to  depict  areas  underground  may  be  made  with 
surveyors'  compass  and  chain,  but  generally  now  the  transit  or 
theodolite  is  used  to  take  the  angles,  and  the  steel  tape  to  meas- 
ure the  distances,  and  in  some  mines  the  tape  may  be  with  ad- 
vantage hundreds  of  feet  in  length ;  but  generally  50  feet  for 
the  chain  or  100  feet  for  the  tape  are  most  convenient  lengths. 

448.  The  surveyor  and  each  assistant,  of  course,  requires  a 
lamp,  and  "the  sights"  are  ranged  with  lamp  and  plummet, 
the  sight  from  the  instrument  being  taken  upon  the  flame  of 
the  miner's  lamp   (or  candle,  it  may  be)  suitably  held  at  the 
plummet  line,  which  is  held  to  depend  from  a  point  fixed  or  to 
be  fixed  in  the  "  roof"  or  over  a  point  in  the  "  bottom."     The 
plummet  string  itself  may  be  seen  within  300  feet.     A  chain- 
pin   (arrow)   can  be  used  to  plumb  the  light  over  or  under  a 
point.     It  is  advised  to  display  the  light  at  a  station  for  sight 
only,  and  therefore  in  moving  it,  for  any  reason,  other  than 
vertically,  in  giving  the  point,  it  should  be    hidden  from  the 
observer. 


MINE   SURVEYING.  381 

The  point  may  be  marked  by  a  nail  in  the  timber  cap  or  sill, 
or  be  a  nail  in  a  peg ;  the  place  of  the  point  in  smooth  roof  is 
to  be  made  conspicuous  by  a  ring  of  white  paint  around  it,  and 
also  as  it  may  be  by  reference  marks  at  the  sides  (pillars)  of 
the  passage-way. 

It  is  a  refinement  to  use  a  lamp  which  is  also  a  plummet,  and 
further  to  place  an  extra  lamp  on  the  bottom  under  it;  two 
tights  seen  in  the  vertical  line  making  its  place  more  certain, 
and  helping  to  decide  that  the  sight  is  ready  to  be  taken.* 

It  may  happen  that  the  line  of  reflection  from  standing  water 
can  be  taken  for  the  line  of  incidence  of  a  light  held  under  a 
point,  when  the  roof  droops  between,  the  passage  being  "  in 
swamp"  there. 

The  surveyor's  lamp  is  made  entirely  of  brass  or  copper,  so  as 
not  to  affect  the  magnetic  needle  of  the  instrument. 

For  use  in  low  openings  the  tripod  of  the  instrument  must  be 
one  of  short  legs  (an  extra  set  of  shifting  legs  will  answer  the 
purpose),  or  have  extension  legs. 

It  has  been  suggested  to  use  two  extra  tripods,  one  to  set 
up  in  advance,  for  keeping  the  place  of  fore-sight  and  for  receiv- 
ing the  instrument,  alone  carried  forward  to  be  mounted  there 
at  the  same  exact  spot  with  facility,  while  the  tripod,  left 
standing  at  the  last  place  of  the  instrument,  marks  the  point 
for  back-sight  with  equal  certainty  :  thus  each  of  three  tripods 
taking  its  turn  in  being  at  a  place  for  fore-sight,  remaining 
there  for  mounting  the  instrument  upon  it,  and  still  remaining 
for  back-sight  after  the  instrument  is  taken  for  mounting  at 
next  station.  There  are  obvious  objections  to  this  in  the 
weight  of  the  luggage,  and  that  only  some  instruments  are 
made  for  such  ready  separate  handling. 

Some  rays  of  light  must  be  thrown  into  the  telescope  at  its 
object  end  to  make  visible  the  cross-hairs  therein.  This  is  gen- 
erally done  by  the  surveyor,  while  taking  a  sight,  holding  his 

*  Eckley  B.  Coxe  derised  the  plummet  lamp,  and  also  a  form  of  it  with 
wire-gauze  covering,  like  the  Davy  Safety  Lamp,  for  use  where  fire-damp 
may  be  expected. 


382  PLANE   SURVEYING. 

lamp  in  his  left  hand  at  the  front,  but  a  little  to  one  side  of  the 
object-glass.  A  reflector  mounted  at  the  object  end  is  a  help. 
One  is  a  silvered  flat  ring,  standing  bias,  about  "2  inches  forward 
from  a  collar  which  is  slipped  over  the  object  end  of  the  tele- 
scope. It  reflects  light  into  the  instrument  as  an  annular  beam. 
Another  one  is  a  diminutive  hemisphere  which  scatters  light 
caught  from  the  lamp  into  the  tube. 

The  change  to,  and  the  equable  temperature  of,  the  mine 
require  the  trying  and  favor  the  making  of  the  ordinary  adjust- 
ments of  the  instrument  there. 

449.  Stations  are  generally  made  only  at  the  angle  points  of 
survey  lines,  and  are  therefore  not  regularly  distanced.     They 
may  be  numbered,  lettered,  or  designated  by  the  total  distance 
from  the  zero  of  the  measurements  of  their  line.     Intermediate 
points  are  made  on  the  line  where,  opposite  to  lateral  openings, 
other  lines  of  survey  or  important  short  connections  by  measure- 
ment merely  may  start.     The  corners  of  chambers  along  the 
passage  may  be  noted  by  distance  without  making  points  ;  the 
size  and  position  of  parts  of  chambers  being  afterwards  taken 
and  noted  by  sketch  with  dimensions  relatively  marked  thereon, 
there  being  mostly  a  parallelism  in  the  rock  measures  which  sim- 
plifies the  position  and  shape  that  chambers  take,  so  that  no 
special  survey  of  directions  is  regularly  required  for  them. 

450.  Angles  between  vertical  planes  of  sight  (in  azimuth) 
are  noted  for  obtaining  the  courses  as  reduced  courses  from  the 
initial  course  of  survey,  by  the  successive  additions  and  sub- 
tractions to  it  and  from  it  of  the  angles  as  taken,  and  modified 
according  to  the  series  of  90°  in  each  quadrant  of  the  circle. 

The  initial  course  had  better  be  referred  to  true  meridian,  and 
comparison  with  bearings  made  with  allowance  for  the  variation 
(declination)  of  the  needle.  But  it  has  always  been  recognized 
that  the  course,  in  degrees  and  minutes,  of  a  quadrant  —  and 
therefore  liable  to  mistakes  as  to  the  particular  one  of  four 
quadrants  —  would  be  absolute  if  the  full  circle  be  graduated 


MINE   SURVEYING.  383 

around  to  90°,  180°,  270°,  and  360°,  in  the  successive  quadrants. 
While  it  is  not  agreed  whether  north  or  south  shall  be  the  zero, 
the  direction  of  graduation  with  the  movement  of  the  hands  on 
the  dial  of  a  watch  or  clock  is  conventionally  fixed.  The  bear- 
ings will  be  a  key  to  which  zero  was  used  in  the  notes. 

451.  It  is  but  seldom  that  in  drifts  of  mines  the  alignment  as 
well  as  the  grade  requires  adjustment  to  the  regularity  of  straight 
lines  and  curves  similar  to  surface  railroads  ;  for  the  tram-cars 
will  run  around  very,  sharp  turns,  and  for  them  there  is  there- 
fore no  necessity  of  expensive  improvements  in  line.  But  when 
a  locomotive  is  to  be  used,  or  wire-rope  haulage  is  to  be  intro- 
duced, there  is  apt  to  be  a  call  for  regulation  of  the  line,  with 
regard,  especially,  to  minimum  radius  of  curvature. 

Unlike  the  longer,  flat  curves  of  a  railroad,  —  designated 
according  to  the  American  system  by  the  even  angular  deflec- 
tions from  each  other  of  chords  of  100  feet,  —  these  sharper 
curves  will  go  by  assumed  even  radii  (in  length  not  less  than 
ten  times  the  gauge  of  track),  and  the  deflection  angles  for  run- 
ning them  in  by  the  instrument  upon  short  chords  will  have  to 
be  calculated. 

One-half  the  chord  divided  by  the  radius  will  equal  the  sine  of 
the  angle  of  deflection  from  tangent,  which  is  half  the  angle 
that  two  such  equal  chords  will  make  with  each  other,  and  also 
half  the  angle  at  the  centre  of  the  circle  subtended  by  the  chord. 
From  any  point  on  the  circular  curve  as  a  position  of  the  instru- 
ment, successive  deflections  of  the  angle  will  fix  the  ends  of 
consecutive  chords  as  measured  in.  Shorter  chords  (like  those 
less  than  100  feet  in  a  railroad  curve)  have  deflection  angles 
approximately  proportional  to  their  lengths. 

For  ranging  the  line  of  direction  of  a  passage  that  is  being 
opened  into  the  solid,  two  points  for  placing  lights  are  given  at 
the  start,  necessarily  near  together,  until  the  prolongation  of 
open  space  allows  testing  the  line  by  the  instrument  and  giving 
new  points  of  line.  From  the  three  points  of  a  curve  line  that 
mark  the  chords  of  half  the  arc,  obviously,  by  simple  measure- 


384  PLANE   SURVEYING. 

ments,  a  like  fourth  point  may  be  derived  as  the  face  (breast) 
of  the  working  is  advanced.  In  driving  a  passage-way  describ- 
ing a  semicircle  —  to  save  weakening  pillar  at  foot  of  shaft  —  a 
long,  curved  gas-pipe  was  used  in  ranging  around.  A  large- 
scale  working  plot  showing  offsets  secures  the  proper  location 
of  curving  and  branching  passages. 

Outside,  besides  the  fixing  of  projected  curves  by  deflection 
angles  as  above,  the  laying  off  of  points  of  arc  intermediate  on 
the  chord  is  by  foot-rule  measurement  of  ordinates  at  right 
angles. 

But  without  strict  regard  to  data,  an  expedient  way  of  uniting 
two  intersecting  straight  lines  of  track  03*  a  circular  curve  (as 
an  arc  starting  from  the  one  straight  line  at  any  distance  short 
of  the  apex  of  the  lines  and  ending  on  the  other  line  an  equal 
distance  from  the  apex)  is  to  find  points  by  linear  measure- 
ment merely.  Assuming  any  tangential  distance  back  from 
apex  to  P.C.  (point  of  curve),  the  beginning,  and  the  same  to 
P.T.  (point  of  tangent),  the  end  of  curve,  we  find  a  third  point 
of  the  arc,  its  middle,  as  a  point  midway  between  the  middle  of 
the  chord  of  the  whole  arc  and  the  apex.  One-fourth  of  this 
versed  sine  will  be  the  versed  sine  (middle  ordinate)  to  be  erected 
on  each  chord  of  half  the  arc  for  points  of  the  arc.  And  any 
other  middle  ordinates  will  be  as  the  squares  of  their  arcs  or 
chords. 

This  principle  applies  in  rounding  off  intersecting  grades 
into  vertical  curves,  either  convex  or  concave  ;  by  vertical  allow- 
ances and  according  to  horizontal  distances,  starting  with  that 
at  the  apex  and  proceeding  similarly  to  the  foregoing  as  to 
subdivisions. 

The  laying  off  of  curves  by  chords  and  versed  sine  so  derived 
does  not  require  knowledge  of  length  of  radius  or  of  amplitude 
of  angle.  But  when  the  extent  of  circular  arc  between  two 
tangents  is  to  be  determined  by  the  length  of  radius,  the  tan- 
gential distance  from  apex  will  equal  radius  multiplied  by  natural 
tangent  of  half  the  angle  of  intersection  ;  and  between  P.C. 
and  P.T.  there  will  be  the  same  measures  of  chord  as  there  are 
of  chord  singles  in  angle  of  intersection. 


MINE   SURVEYING.  385 

452.  In  the  note-book  the  left-hand  page  is  used  for  stations, 
distances,  angles,  courses  (reduced),  and  bearings  (magnetic), 
and  the  opposite  right-hand  page  for  offset  distances  —  marked 
relative  to  a  perpendicular  line  dividing  the  page,  together  with 
sketches  and  remarks.  The  notes  should  begin  at  the  bottom 
of  the  pages  and  proceed  upwards,  to  appear  as  on  the  plan  to 
which  their  results  are  to  be  transferred,  in  their  proper  relation 
of  position  and  observation  forward. 

The  plan  of  underground  work  is  begun  with  the  plotted  net- 
work of  the  lines  of  survey,  then  the  outline  of  parts  excavated 
is  drawn  in  detail,  and  these  are  shaded,  as  the  places  become 
closed  in  and  abandoned,  to  distinguish  what  is  open  work  at 
any  period. 

The  scale  of  maps  showing  the  workings,  etc.,  of  coal  mines 
is  now  fixed  by  law  in  many  of  the  States  at  1  :  1200  as  the 
least;  that  is,  at  not  less  than  1  inch  for  100  feet;  the  purpose 
of  the  maps  being  to  aid  the  official  inspection  and  regulation 
of  the  mines  for  securing  the  health  and  safety  of  the  miners. 
The  plan  will  generally  require  to  show  the  relation  of  the 
workings  to  surface  openings,  watercourses,  and  bounding  lines, 
and  to  improvements,  such  as  buildings,  roads,  and  railroads. 

The  line  of  outcrops  (exposure  at  the  surface  of  the  ground  of 
the  mineral  beds)  within  its  range  will  appear  on  the  map,  but 
general  topographical  detail  is  reserved  for  the  extended  small- 
scale  maps  of  the  surface,  which  will  represent  what  may  be 
learned  of  mineral  indications  also ;  from  which  data  in 
advance  of  the  workings  may  be  derived  and  confirmed 
by  special  explorations,  as  of  proof-holes  and  deep  boring. 
But  upon  the  mine  plaii  such  elevations  (heights  of  surface 
above  datum)  as  seem  most  essential,  such  as  principal  ones 
along  the  outcrops,  highest  points  of  hills,  and  lowest  of 
streams  should  be  mapped. 

The  use  of  the  pantograph,  for  reducing  the  irregular  figures 
of  mine  plans  with  all  details  from  one  scale  to  another,  has 
found  much  approval ;  and  the  plammeter  is  liked  for  labor- 
saving  and  accuracy  in  determining  such  areas. 


886  PLANE   SURVEYING. 

453.  In  veins,  the  work  being  deep  and  narrow,  and  pursued 
from  levels  or  galleries  (horizons  of  working)  generally  about 
60  feet  apart  in  height,  plans  of  these  levels,  drawn  in  different 
colors  to  distinguish  them,   are  superimposed  on  the  map  of 
general  plan.      The}'  show  the  openings,  —  the  gangways,  the 
cross-cuts,  etc.,  —  with  the  defining  lines  of  the  walls  of  the  vein, 
and  may  embrace  other  separations  of  the  mineral.     Longi- 
tudinal   elevation    and   vertical   cross-sections    will    show   the 
shafts  and  other  connections  between  the  levels,  together  with 
the  chambers,  whether  open,  filled  in,  or  caved. 

Ore  bodies  occurring  detached  and  of  the  most  varying 
dimensions,  though  often  resembling  each  other  as  lenticular  in 
shape,  make  the  workings  appear  in  plan,  elevation,  and  cross- 
section,  as  the  results  of  exploration  in  patches.  Shafts  in  the 
vein  will  be  parallel  to  pitch  of  one  wall,  and  therefore  varying 
from  the  vertical. 

A  stratified  bed  that  is  to  be  operated  upon, — opened,  and 
won  by  mining,  —  may  be  conceived  as  a  seam  of  uniform  small 
thickness  extending  within  limits  as  a  plane  surface  and  in 
relative  position  defined  by  the  "  strike  "  (the  course  of  all  its 
level  lines,  which  will  all  be  parallel)  and  its  "dip"  (the 
greatest  pitch  at  right  angles  to  the  course  of  the  level-line). 
But  upon  the  large  scale  the  seam  occurs  of  variable  thickness, 
and  with  lines  of  level  changing  in  direction  and  not  parallel  at 
different  elevations,  to  the  degree  that  instead  of  a  plane  it  is  a 
warped  surface. 

The  arrangement  of  permanent  works  upon  the  surface  of  the 
ground  with  reference  to  the  lay  of  the  bed  as  well  as  the  topog- 
raphy and  improvements  existing  or  suited  to  it,  the  favorable 
connection  of  the  lines  of  haulage  and  drainage  inside,  with  all 
to  govern  outside,  present  to  the  mind  of  the  mathematical 
surveyor  applications  of  the  theorems  of  Descriptive  Geometry, 
as  included  in  adaptation  to  the  ends  of  practical  economy. 

454.  Location  upon  the  surface  of  the  ground  of  the  plan  of 
inside  work,  is  a  repetition  of  courses  and  distances  outside  in  the 


MINE   SURVEYING.  387 

same  vertical  planes.  Any  particular  portion  of  the  workings 
in  progress  can  thus  be  compared  in  natural  scale  upon  actual 
plan  of  surface  of  the  ground  over  them. 

Overlaid  plans  with  elevations  and  cross-sections  of  workings, 
such  as  were  described  for  workings  in  veins,  are  required  to 
show  the  development  in  high  pitching  beds.  The  "  lifts  "  or 
levels  in  such  of  coal  are  100  yards  apart,  measured  on  line  of 
pitch. 

Overlaid  plans  of  different  parallel  seams  worked  through 
same  shaft  are  also  made,  but  without  systematic  elevation 
and  cross-section  ;  the  connections  (shafts,  slopes,  or  tunnels) 
between  the  beds  being  through  barren  ground,  and  limited 
to  the  exigencies  of  hoisting,  draining,  and  ventilating. 

455.  Following  the  determination  in  azimuth  by  courses  and 
distances  of  the  passages  in  the  mine  is  the  determination  of 
their  changes  in  level  by  the  spirit  levelling-instrument  and  the 
level-rod  (as  a  separate  operation,  even  if  the  transit  be  a  com- 
bined instrument  having  a  parallel  spirit  level  attached  to  its 
telescope),  the  work  being  quite  similar  to  such  above  ground. 
But  the  rod  must  be  limited  in  height  to  the  low  spaces  where  it 
is  to  be  used,  and  is  preferably  marked  with  red  figures  for  the 
feet,  and  white  figures  for  the  tenths,   upon  a  black  ground. 
The  top  of  a  simple  white  target  is  safer  to  take,  however,  than 
the  reading  from  the  instrument  of  the  figures  themselves.     For 
accuracy,  sights,  as  above  ground,  should  be  limited  to  300 
feet  in  distance  from  the  instrument. 

From  the  elevations  of  points  taken  by  levelling,  contour 
lines  can  be  shown  on  plan  as  the  mineral  bed  is  exploited. 

Blue  is  the  conventional  color  for  these  contour  lines  and  the 
figures  marking  their  elevation  above  the  datum,  on  a  mine 
plan,  and  brown  suits  for  the  contrasted  surface  elevations. 

456.  Levelling  along  passage-ways  for  the  purpose  of  fixing 
better  gradients  of  hauling-roads,  or  for  fall  of  water  by  rectifi- 
cation of  undulating  bottom  to  improve  drainage,  requires  sta- 


388  PLANE    SURVEYING. 

tions  especially  chained  in  at  regular  distances  of  50  feet  or 
less  ;  the  marks  being  temporary  ones  on  the  sides  to  serve  for 
taking  the  levels  and  to  be  referred  to  as  to  heights  in  grading, 
when  the  variation  of  level  of  bottom  from  the  grade  of  a 
station  governs  the  cutting  or  filling  of  bottom  there,  or  change 
of  the  whole  cross-section  in  height,  as  it  may  be.  For  the 
adoption  of  suitable  gradients  along  an  extended  line,  a  longi- 
tudinal vertical  section  is  drawn,  called  a  profile,  which  exhibits 
the  relation  of  ground-line  levels,  and  allows  the  fixing  of 
grade  with  assurance.  The  profile  may  include  the  line  of  top 
as  well  as  of  bottom,  with  section  of  rock  measures  to  be 
affected  by  "•  ripping  "  of  the  roof  and  "  cutting  "  of  bottom. 

457.  A  Drift  or  passage  along  with  the  measures  of  a  bed 
will  make  undulating  grade,  if  course  be  followed  ;  and  if  the 
drainage-rise  be  allowed  to  govern,  the  alignment  will  be  sacri- 
ficed. 

Tunnelling,  however,  being  arbitrary,  across  the  measures,  is 
mostly  upon  directed  line  and  grade.  Slopes  are  mostly  upon 
directed  course ;  but  if  within  the  measures  of  an  inclined  bed 
will  mostly  be  variable  in  grade.  So  with  an  adit,  driven  to 
give  drainage  outfall  to  the  surface.  For  it,  shortening  of  the 
distance  will  probably  be  the  governing  condition  principally. 

458.  For  the  workings  at  high  pitch,  the  determination  of 
horizontal   and    vertical   components   of  the   distances  on  the 
sloph  g  lines  of  top  and  bottom  in  a  bed,  and  "  hanging  wall " 
and  "foot  wall"  in  a  vein,  will  bring  the  vertical  arc  of  the 
instrument  into  requisition,  for  obtaining  the  vertical   angle, 
which  is  always  taken  as  the  full  angle  above  the  horizontal. 
Vertical   sections,    besides   such    longitudinal    ones    following 
broken  line  of  passage  within  a  stratum  and  showing  only  adja- 
cent rock  measures,  may  be  made  of  particular  places  where 
there  is  folding,  or  fault,  of  the  measures,  and  for  geological  or 
more  general  purposes  they  may  exhibit  the  lay  and  thickness 
of  the  various  rocks  up  to  the  surface,  which  will  as  a  correct 


MINE   SURVEYING.  389 

margin  show  the  outcroppings  in  profile.  Vertical  sections  may 
be  projections  upon  planes  that  traverse  the  measures  according 
to  various  conditions,  and  may  be  constructed  of  related  points 
from  the  map  that  were  not  determined  for  their  relevancy  to 
this  purpose. 

It  seems  that  vertical  arcs  have  had  versed  sines  correspond- 
ing to  radius  1  marked  around  them  for  the  purpose  of  telling 
the  allowance  upon  slope  measurement  to  obtain  corresponding 
horizontal  distances,  the  versed  sine  being  the  difference 
between  the  hypothenuse  as  the  radius  and  the  horizontal  base 
as  the  cosine  of  the  vertical  right-angled  triangle  formed  ;  and 
the  slope  length  for  a  given  horizontal  distance  would  be 
greater,  according  to  the  versed  sine  of  the  angle. 

Vertical  arcs  have  had  tangents  as  rises  corresponding  to  the 
unit  of  horizontal  distance  for  the  different  angles  marked  upon 
them. 

A  method  of  dividing  the  arc  according  to  the  sines,  without 
the  intervention  of  the  equal  graduation  into  degrees  neces- 
sarily, is  the  subject  of  a  contribution  to  "Van  Nostrand's 
Engineering  Magazine"  for  July,  1876,  and  is  appended  at  the 
end  of  this  chapter. 

459.  The  measurement  down  deep  borings  or  shafts  is  best 
made  by  special  flat  steel  wire,  with  suitable  plummet  heavy 
enough  to  insure  its  making  the  wire  line  taut. 

The  transfer  of  points  down  a  shaft,  as  of  two  to  determine 
a  base  line  for  connecting  surveys  below  with  those  on  the 
surface  of  the  ground,  is  made  by  very  heavy  plummets 
attached  to  ordinary  wire  run  off  of  reels.  A  portable  box 
to  contain  the  reels,  their  cranks,  and  the  plummets,  is  con- 
venient; the  best  arrangement  being  that  of  reels  fixed  in  a 
frame  that  stays  in  the  box.  The  suspended  plummets  are  to 
be  received  below  each  in  a  bucket  of  water,  or,  if  hanging  from 
considerable  height,  in  some  thicker  liquid  to  settle  the  wire 
lines  to  a  steady  position  for  ranged  observation  by  the  instru- 
ment below.  And  the  observation  will  be  easier  upon  wire  that 
is  whitened  there  by  chalk  or  paint  after  being  placed. 


390  PLANE   SURVEYING. 

The  plummets  in  the  shaft  of  the  Washington  Monument,  for 
showing  changes  in  the  verticality  of  the  structure,  are  steadied 
in  vessels  containing  a  mixture  of  glycerine  and  molasses. 

460.  For  taking  courses  on  pitches  at  high  angles  an  extra 
telescope  on  the  axis  extended  to  the  outside  of  one  of  the 
standards  of  transit  has  been  used.     Another  mining  transit 
has  for  the  same  purpose  the  sweep  of  the  telescope  to  the  ver- 
tical  position,  made   possible    by   having   its   standards  made 
inclined  to  overhang.     But  the  object-prism  placed  before  the 
object-glass,  allowing  sighting  at  true  right  angles  in  any  plane, 
seems  most  simply  to  fulfil  the  requirements  for  sighting  up  or 
down,  as  well  as  side  wise,  and  is  a  ready  means  applicable  to 
the  telescope  of  any  ordinary  instrument.     A  transit  adapted 
in  any  of  these  ways   for  taking  vertical  sights  enables  the 
points  of  base  line,  as  transferred  by  plummets  to  the  bottom  of 
the  shaft,  to  be  tested  and  compared  with  the  extended  line 
across  the  pit  top,  provided  the  atmosphere  be  clear  in  the  shaft 
and  obstructions  do  not  intervene.     The  vertical  adjustment 
of  the  instrument  itself  would  be  tested  by  this  check,  the  usual 
test  being  on  high  objects,  with  reversal  of  standards  to  oppo- 
site sides  by  turning  the  horizontal  plates. 

A  heavy,  substantial,  simple  transit,  not  weighted  with 
"attachments,"  is  the  most  reliable. 

461.  The  use  of  the  hanging  compass  and  of  the  hanging 
clinometer  of  the  olden  time  is  retained  in  small  and  crooked 
passages  of  some  metalliferous  mines.     And  their  subsidiary 
use  in  excavations  inconvenient  of   access  or  footing  of   the 
ordinary    (the   standing   instruments)    has  lately  been   recom- 
mended as  of  wider  application,  and  they  have  been  introduced 
into  this  country.     Each  of  the  instruments  is  to  hang  by  its 
two  hooks,  turned  opposite  ways,  to  the  cord  that  marks  the 
line.     The   compass-box  levels  itself  by  its 'gimbals    (double 
trunnions),  like  a  ship's  compass,  in  the  frame  of  which  the 
flat  hooks  with  long  bearings  in  line  are  a  part.     The  clinometer 


MINE   SURVEYING.  391 

bangs  as  a  vertical  arc  with  plummet  to  give  the  inclination  of 
the  cord  from  the  horizon,  while  the  compass  gives  the  needle 
course.  The  cord  is  stretched  from  one  low  stout  tripod  to 
another,  or  in  a  curving  space  may  be  fastened  to  a  gimlet 
screwed  into  side  timber  beyond  intersecting  point  or  angle  of 
two  cords.  The  tripod  serves  as  a  stool  also  for  the  assistant 
holding  cord  to  the  point  on  it  firmly.  The  distances  are  accu- 
rately measured  along  the  cord  by  applying  a  graduated  rod  to 
it.  The  horizontal  and  vertical  components  of  the  measure- 
ments have  to  be  calculated  for  plotting  on  plan  and  section. 
In  the  old  mining  regions  of  Europe  the  surface  surveys  were 
also  carried  on  with  the  same  appliances.  With  .care  and 
patience  surprisingly  good  results  in  locating  connections  were 
attained.  The  old  instruments  were  graduated  in  hours  and 
minutes,  and  the  English  designations  of  dial  and  dialling  for 
the  mine  compass  and  operations  with  it  seem  to  refer  to  the 
same  original  division  of  its  circle.  It  seems  strange  to  learn 
that  the  plotting  was  protracted  by  the  same  compass  (swung 
there  on  horizontal  plate  used  for  straight  edge),  reference 
being  had  to  a  meridian  line  fixed  in  the  office,  and  the  drawing- 
table  being  a  smooth  and  level  stone  slab  resting  on  foundation 
independent  of  the  office  floor. 

462.  Formerly,  when  topography  was  used  more  for  the  pic- 
turing of  the  plan  of  landscape  in  mapping  the  features  for  the 
information  of  the  tourist  or  the  military  commander,  than  for 
the  projection  of  the  contour  accurately  to  fit  the  location  of 
artificial  ways  of  the  different  kinds  to  the  ground,  hachures 
were  used  to  indicate  character  of  sloping  elevations,  and 
they  survive  in  use  upon  small-scale  maps,  to  indicate  moun- 
tain chains.  They  are  intended  to  be  lines  of  pitch,  drawn 
close  together  so  as  to  graduate  changes  naturally,  and  they 
should  be  broken  at  the  intersection  of  the  successive  level 
planes  with  the  surface  to  make  terraces  however  narrow,  and 
suggest  level  stages  in  measure  of  elevation.  Now  we  have  on 
topographical  plans  contour  lines  to  represent  the  lines  of  sue- 


392  PLANE   SURVEYING. 

cessive  levels,  say  10  feet  apart  in  rise.  They  are  plotted  by 
connecting  all  points  of  elevation  that  may  be  determined  over 
the  area  with  regard  to  the  requirements  of  accuracy  in  noting 
the  changes ;  and  they  may  be  considered  the  margins  made  by 
a  body  of  water  that  had  successively  risen  or  receded  10  feet 
in  height  at  a  time  over  the  area.  They  are  to  be  marked  by 
their  elevation  above  the  lowest  datum  plane,  preferably  over 
that  of  mean  tide  of  the  ocean.  They  turn  upon  themselves 
where  they  enclose  a  peak  or  a  basin  —  according  as  the  next 
ones  indicate  them  as  higher  or  lower  in  the  series  ;  they  are 
farther  apart  in  horizontal  distance  as  slopes  are  flatter,  and 
where  two  or  more  coincide  for  any  distance  there  is  a  precipice. 

These  points  of  even  elevations  of  the  ground  are  determined 
from  the  levels  run  along  the  survey  lines,  and  the  cross-section 
profiles  taken  at  the  stations  of  the  lines — slopes  being  taken  at 
right  angles  to  the  line  with  straight  edge  pole  and  clinometer  or 
plummet  slope  level  applied  to  it.  Each  of  these  angle  instru- 
ments having  a  vertical  graduated  arc,  the  former  with  arm 
hinged  at  centre  of  arc  and  carrying  a  spirit-level  to  ascertain 
the  vertical  angle  included  between  the  levelled  arm  and  the 
slope  of  the  straight  edge  under  it ;  the  latter,  by  the  departure 
from  the  perpendicular  of  the  plummet,  showing  the  equal 
departure  from  the  horizontal  of  the  straight  edge. 

From  the  profile  of  each  slope  sketched  in  the  field-book  and 
marked  with  distances  and  degrees  of  rise  and  fall  across  the 
survey  line,  the  successive  even  10-foot  points  can  be  laid  off  on 
plan,  regard  being  had  in  starting  with  elevation  of  station  to 
the  partial  changes  required  for  the  first  even  10-foot  point  each 
way.  A  scale  of  horizontal  distances  for  each  degree  of  the 
arc,  to  gain  10  feet  rise,  is  made  by  the  topographer  of  Bristol- 
board  to  lay  off  the  points  derived  by  sloping  at  the  stations, 
and  saves  the  plotting  of  the  profile  of  cross-section. 

The  topographer  prefers  to  draw  the  contours  in  the  field  as 
taken,  using  demi-sheets  of  paper  that  can  be  joined  at  their 
margins,  and  upon  each  of  which  a  portion  of  the  line  corre- 
sponding to  its  number  is  plotted,  the  line  having  dots  along  it, 
spacing  the  successive  stations  intermediate  of  the  angle  points 


MINE   SURVEYING.  393 

of  line,  and  having  the  elevations  corresponding  in  pencil  along- 
side. The  sheets  are  held  in  a  box  that  is  carried  by  a  shoulder- 
strap,  and  the  side  of  which  is  used  in  the  field  as  a  drawing- 
board,  the  particular  sheet  in  use  at  the  time  being  tacked  on  it. 

463.  The  topographer  will  sketch  in  the  streams,  buildings, 
etc.,  with  reference  to  measurements  however,  and  will  have 
special  lines  with  small  compass,  etc.,  run  for  him  to  make 
contour  connections.  The  operations  will  rise  to  the  scope  of 
plane-table  work,  if  the  drawing-board  have  a  socket  with 
clamps,  and  be  mounted  and  levelled  (by  applying  a  loose  hand- 
level)  upon  a  tripod  ;  the  ruler  used  on  it  having  small  compass 
sights  screwed  to  its  ends  for  sighting  to  objects  and  fixing 
their  position  on  plot  by  the  graphic  triangulation  of  intersected 
sight-lines  from  different  stations  on  the  survey -line ;  the  sta- 
tion on  plot  when  over  its  place  on  ground  having  a  needle 
stuck  upright  in  it,  that  has  a  sealing-wax  head  for  convenient 
handling,  for  the  purpose  of  resting  the  ruler  against  when  sight- 
ing. Interpolation,  or  resection,  is  the  reverse  sighting  from 
without  the  line  over  the  plot  to  two  or  three  poles  on  stations 
of  the  line  or  other  previously  located  objects,  to  attain  posi- 
tion, it  being  understood  that  the  plane  table  stands  with  plot 
in  proper  relative  position  always.  Secondary  triangulation  will 
extend  the  area  of  topographic  sketching,  but  this  should  be 
checked  by  connections  beyond  with  surveyed  lines  and  levels. 

The  Locke  level  may  be  used  for  taking  rises  by  finding  all 
the  points  in  sight  that  are  at  a  level  of  the  eye,  and,  in  con- 
nection with  the  le veiling-rod,  the  fall  of  ground  may  also  be 
determined  by  this  instrument.  For  gently  undulating  ground 
the  use  of  it  is  better  than  sloping. 

464.  Contour  lines  are  drawn  10  feet  apart  in  elevation  on 
most  plans  of  extended  land  and  other  surveys  that  are  meas- 
ured in  detail,  but  it  is  obvious  that  cases  occur  where  for  large- 
scale  work  they  are  taken  closer  in  elevation  or  farther  for 
small-scale  mapping.  In  the  fonm-r  rase  of  large-scale  work 
they  may  be  required  exactly  as  elevations  directly  located  by 


394  PLANE   SURVEYING. 

spirit  levelling-instrument,  in  the  latter  case  as  the  approxima- 
tion from  altitudes  taken  in  a  few  places  by  the  barometer. 

The  scope  of  their  usefulness  on  plans  for  projecting  improve- 
ments it  would  be  difficult  to  describe  exhaustively.  They  may 
be  for  use  in  locating  the  drives  and  walks  and  terraces,  etc., 
of  a  park;  the  shaping  of  grounds,  under-draining,  etc.,  about 
a  residence  ;  the  laying  out  of  streets,  etc.,  in  a  hilly  town  ;  the 
leading  of  streams  of  water,  large  or  small,  for  all  purposes  in 
partial  or  wholly  artificial  channels,  for  navigation,  water 
power  and  supply,  irrigation,  etc.  ;  the  location  of  roads  and 
railroads  with  regard  to  ease  of  construction  and  of  favorable 
gradients,  as  well  as  the  uses  in  mining  directly,  and  location 
of  all  surface  erections  collateral  thereto  or  elsewhere,  collec- 
tively known  as  "  the  Works." 


ANGULAR  CROSS-SECTIONING. 

By  F.  Z.  SCHELLENBERG,  C.E. 

Written  for  "Van  Nostrand's  Engineering  Magazine,"  July,  1876. 

A  most  direct  and  expeditious  method  to  get  differences  in 
level  between  points  in  sight  is  by  the  use  of  a  vertical  arc  grad- 
uated to  the  successive  sines  1,  2,  3,  ...  100,  in  quadrant,  for 
the  radius  of  arc  100. 

Multiplying  the  distance  measured  in  hundreds  on  the  slope 
by  the  rate  per  hundred  indicated  on  the  arc  gives  the  difference 
in  level  in  units.  In  the  higher  parts  of  the  arc  the  correspond- 
ing cosines  may  be  marked  for  deriving  horizontal  distances. 

The  applicability  of  this  graduation  to  such  purposes,  as 
described  under  this  caption  by  R.  Bell,  C.E.,  in  May  number, 
is  obvious,  as  may  also  be  its  use  for  more  extended  profiles, 
for  geological  cross-sections,  for  road-grading,  or  wherever 
between  points  obtained  by  the  levelling-instrument  its  accuracy 
is  not  indispensable. 

A  clinometer  thus  graduated  enables  contour  lines  for  topo- 
graphical work  to  be  most  readily  determined.  The  table 
following  gives  the  100  points  in  the  quadrant  in  terms  of  the 
common  graduation  of  90°  to  the  quadrant. 


MINE    SURVEYING. 


395 


Vertical 

Horizontal 

Vertical 

Horizontal 

Distance  for 

Distance  for 

Angle 

Distance  for 

Distance  for 

Angle 

100  measured 

100  measured 

with  Horizon. 

100  measured 

100  measured 

with  Horizon. 

on  Slope. 

on  Slope. 

on  Slope. 

on  Slope. 

0 

0°00' 

51 

:;u  'in' 

1 

.  ... 

0°34' 

52 

... 

31°  20' 

2 



1°09' 

53 

..  .  . 

32°  00' 

3 

.  .'.. 

1°43' 

54 

32°  41' 

4 

• 

2°  18' 

55 

88.6 

33°  22' 

6 

99.9 

2°  52' 

56 

34°  03' 

6 

.... 

3°  26' 

57 

34°  45' 

7 



4°  01' 

58 

35°  27' 

8 



4°  35' 

59 

36°  09' 

9 

5°  10' 

60 

80.6 

36°  52' 

10 

99.6 

5°  44' 

61 

.... 

37°  35' 

11 

6°  19' 

62 

38°  19' 

12 

6°  54' 

63 

.... 

39°  03' 

13 

7°  28' 

64 

39°  48' 

14 

8°  03' 

65 

76.0 

40°  32' 

15 

98.9 

8°  38' 

66 



41°  18' 

16 

9°  12' 

67 

.... 

42°  04' 

17 

9°  47' 

68 

42°  51' 

18 

10°  22' 

69 

43°  38' 

19 

10°  57' 

70 

71.4 

44°  26' 

20 

98.6 

11°  32' 

71 

45°  14' 

21 

12°  07' 

72 

46°  03' 

22 

12°  43' 

73 

46°  53' 

23 

13°  18' 

74 

47°  44' 

24 

13°  53' 

75 

66.2 

48°  35' 

25 

96.8 

14°  29' 

76 

49°  28' 

26 

15°  04' 

77 

50°  21' 

27 

15°  40' 

78 

.... 

61°  16' 

28 

16°  16' 

79 

62°  11' 

29 

16°  51' 

80 

60.6 

63°  08' 

30 

95.4 

17°  27' 

81 

64°  06' 

31 

18°  04' 

82 

66°  05' 

32 

18°  40' 

83 

66°  06' 

33 

19°  16' 

84 

67°  08' 

34 

19°  53' 

85 

52.7 

58°  13' 

35 

93.7 

20°  29' 

86 

59°  197 

36 

21°  06' 

87 

.... 

60°  28' 

37 

21°  43' 

88 

.... 

61°  39' 

38 

22°  207 

89 

62°  62' 

39 

22°  57' 

90 

43.6 

(il     (I!)' 

40 

91J6 

23°  35' 

91 

66°  30' 

41 

24°  12' 

92 

6(5°  66' 

42 

24°  50' 

93 

.... 

68°  26' 

43 

25°  28'' 

94 

70°  03' 

44 

26°  0(5' 

95 

31.2 

71     is' 

45 

89.3 

26°  45' 

9(5 

73°  44' 

46 

27°  23' 

97 

.... 

7:.   :.(»' 

47 

28°  02' 

98 

78°  81' 

48 

28°  41' 

99 

81°  64' 

49 

29°  20' 

100 

00.6 

90°  00' 

50 

86.6 

30°  00' 

TRANSIT, 
A.S  FIRST  MADE  IN  1831  BI  THE  INVENTOR,  WILLIAM  J.  YOUNG,  PHILADELPHIA,  PA 


APPENDIX. 

THE  JUDICIAL  FUNCTIONS   OF  SUEVEYOES* 

A* 

WHEN  a  man  has  had  a  training  in  one  of  the  exact  sciences, 
where  every  problem  within  its  purview  is  supposed  to  be  sus- 
ceptible of  accurate  solution,  he  is  likely  to  be  not  a  little  impa- 
tient when  he  is  told  that,  under  some  circumstances,  he  must 
recognize  inaccuracies,  and  govern  his  actions  by  facts  which 
lead  him  away  from  the  results  which  theoretically  he  ought  to 
reach. 

Observation  warrants  us  in  saying  that  this  remark  may 
frequently  be  made  of  surveyors. 

In  the  State  of  Michigan,  all  our  lands  are  supposed  to  have 
been  surveyed  once  or  more,  and  permanent  monuments  fixed 
to  determine  the  boundaries  of  those  who  should  become  propri- 
etors. The  United  States,  as  original  owner,  caused  them  all 
to  be  surveyed  once  by  sworn  officers,  and  as  the  plan  of  sub- 
division was  simple,  and  was  uniform  over  a  large  extent  of 
territory,  there  should  have  been,  with  due  care,  few  or  no 
mistakes :  and  long  rows  of  monuments  should  have  been 
perfect  guides  to  the  place  of  any  one  that  chanced  to  be  miss- 
ing. The  truth  unfortunately  is,  that  the  lines  were  very  care- 
lessly run,  the  monuments  inaccurately  placed ;  and,  as  the 
recorded  witnesses  to  these  were  many  times  wanting  in  perma- 
nency, it  is  often  the  case  that  when  the  monument  was  not 
correctly  placed,  it  is  impossible  to  determine  by  the  record,  by 
the  aid  of  anything  on  the  ground,  where  it  was  located.  The 
incorrect  record  of  course  becomes  worse  than  useless  when  tin- 
witnesses  it  refers  to  have  disappeared. 

It  is,  perhaps,  generally  supposed  that  our  town  plats  were 


By  Chief  Justice  Cooley  of  tlie  Supreme  Court  of  Michigan. 


398  PLANE    SURVEYING. 

more  accurately  surveyed,  as  indeed  they  should  have  been; 
for  in  general  there  can  have  been  no  difficulty  in  making  them 
sufficiently  perfect  for  all  practical  purposes.  Many  of  them, 
however,  were  laid  out  in  the  woods ;  some  of  them  by  proprie- 
tors themselves,  without  either  chain  or  compass,  and  some  by 
imperfectly  trained  surveyors,  who,  when  land  was  cheap,  did 
not  appreciate  the  importance  of  having  correct  lines  to  deter- 
mine boundaries  when  land  should  become  dear. 

The  fact  probably  is,  that  town  surveys  are  quite  as  inaccurate 
as  those  made  under  authority  of  the  general  government.  It 
is  now  upwards  of  fifty  years  since  a  major  part  of  the  public 
survevs,  in  what  is  now  the  State  of  Michigan,  were  made  under 
authority  of  the  United  States.  Of  the  lands  south  of  Lansing, 
it  is  now  forty  years  since  the  major  part  were  sold  and  the 
work  of  improvement  began.  A  generation  has  passed  away 
since  they  were  converted  into  cultivated  farms,  and  few,  if  any, 
of  the  original  corner  and  quarter  stakes  now  remain. 

The  corner  and  quarter  stakes  were  often  nothing  but  green 
sticks  driven  into  the  ground.  Stones  might  be  put  around  or 
over  these  if  they  were  handy,  but  often  they  were  not,  and  the 
witness  trees  must  have  been  relied  upon  after  the  stake  was 
gone.  Too  often  the  first  settlers  were  careless  in  fixing  their 
lines  with  accuracy  while  monuments  remained,  and  an  irregular 
brush-fence,  or  something  equally  untrustworthy,  may  have  been 
relied  upon  to  keep  in  mind  where  the  blazed  line  once  was.  A 
fire  running  through  this  might  sweep  it  away,  and  if  nothing 
was  substituted  in  its  place,  the  adjoining  proprietors  might  in 
a  few  years  be  found  disputing  over  their  lines,  and  perhaps 
rushing  into  litigation,  as  soon  as  they  had  occasion  to  cultivate 
the  land  along  the  boundary.  If  now  the  disputing  parties  call 
in  a  surveyor,  it  is  not  likely  that  any  one  summoned  would 
doubt  or  question  that  his  duty  was  to  find,  if  possible,  the  place 
of  the  original  stakes  which  determine  the  boundary  line  between 
the  proprietors. 

However  erroneous  may  have  been  the  original  survey,  the 
monuments  that  were  set  must  nevertheless  govern,  even  though 


APPENDIX.  .    399 

the  effect  be  to  make  one  half-quarter  section  90  acres,  and  the 
one  adjoining  70 ;  for  parties  buy,  or  are  supposed  to  buy,  in 
reference  to  these  monuments,  and  are  entitled  to  what  is  within 
their  lines,  and  no  more,  be  it  more  or  less.  While  the  witness 
trees  remain,  there  can  generally  be  no  difficulty  in  determining 
the  locality  of  the  stakes.  When  the  witness  trees  are  gone,  so 
that  there  is  no  longer  record  evidence  of  the  monuments,  it  is 
remarkable  how  many  there  are  who  mistake  altogether  the  duty 
that  now  devolves  upon  the  surveyor.  It  is  by  no  means  uncom- 
mon that  we  find  men,  whose  theoretical  education  is  thought  to 
make  them  experts,  who  think  that  when  the  monuments  are 
gone,  the  only  thing  to  be  done  is  to  place  new  monuments 
where  the  old  ones  should  have  been,  and  would  have  been  if 
placed  correctly.  This  is  a  serious  mistake.  The  problem  is 
now  the  same  that  it  was  before :  To  ascertain  by  the  best 
lights  of  which  the  case  admits  where  the  original  lines  were. 
The  mistake  above  referred  to  is  supposed  to  have  found  expres- 
sion in  our  legislation  ;  though  it  is  possible  that  the  real  intent 
of  the  act  to  which  we  shall  refer  is  not  what  is  commonly  sup- 
posed. An  act  passed  in  1869  (Compiled  Laws,  593),  amend- 
ing the  laws  respecting  the  duties  and  powers  of  county  survey- 
ors, after  providing  for  the  case  of  corners  which  can  be  identi- 
fied by  the  original  field  notes  or  other  unquestionable  testimony, 
directs  as  follows  : 

"Second.  Extinct  interior  section  corners  must  be  re-estab- 
lished at  the  intersection  of  two  right  lines  joining  the  nearest 
known  points  on  the  original  section  lines  east  and  west  and 
north  and  south  of  it. 

"  Third.  Any  extinct  quarter-section  corner,  except  on  frac- 
tional lines,  must  be  re-established  equidistant  and  in  a  right 
line  between  the  section  corners  ;  in  all  other  cases,  at  its  pro- 
portionate distance  between  the  nearest  original  corners  on  the 
same  line." 

.The  corners  thus  determined,  the  surveyors  are  required  to 
perpetuate  by  noting  bearing  troes  when  timber  is  near.  To 


400  PLANE    SURVEYING. 

estimate  properly  this  legislation,  we  must  start  with  the  ad- 
mitted and  unquestionable  fact  that  each  purchaser  from  gov- 
ernment bought  such  land  as  was  within  the  original  boundaries, 
and  unquestionably  owned  it  up  to  the  time  when  the  monuments 
became  extinct. 

If  the  monument  was  set  for  an  interior  section  corner,  but 
did  not  happen  to  be  "at  the  intersection  of  two  right  lines 
joining  the  nearest  known  points  on  the  original  section  lines 
east  and  west  and  north  and  south  of  it,"  it  nevertheless  deter- 
mined the  extent  of  his  possessions,  and  he  gained  or  lost 
according  as  the  mistake  did  or  did  not  favor  him. 

It  will  probably  be  admitted  that  no  man  loses  title  to  his 
land  or  any  part  thereof  merely  because  the  evidences  become 
lost  or  uncertain.  It  may  become  more  difficult  for  him  to  es- 
tablish it  as  against  an  adverse  claimant,  but  theoretically  the 
right  remains  ;  and  it  remains  as  a  potential  fact  so  long  as  he 
can  present  better  evidence  than  any  other  person.  And  it 
may  often  happen  that  notwithstanding  the  loss  of  all  trace  of 
a  section  corner  or  quarter  stake,  there  will  still  be  evidence 
from  which  anv  surveyor  will  be  able  to  determine  with  almost 
absolute  certainty  where  the  original  boundary  was  between  the 
government  subdivisions. 

There  are  two  senses  in  which  the  word  "extinct"  may  be  used 
in  this  connection  :  one,  the  sense  of  physical  disappearance ; 
the  other,  the  sense  of  loss  of  all  reliable  evidence.  If  the 
statute  speaks  of  extinct  corners  in  the  former  sense,  it  is  plain 
that  a  serious  mistake  was  made  in  supposing  that  surveyors 
could  be  clothed  with  authority  to  establish  new  corners  by  an 
arbitrary  rule  in  such  cases.  As  well  might  the  statute  declare 
that  if  a  man  loses  his  deed,  he  shall  lose  his  land  altogether. 
But  if  by  extinct  corner  is  meant  one  in  respect  to  the  actual 
location  of  which  all  reliable  evidence  is  lost,  then  the  following 
remarks  are  pertinent : 

1.  There  would  undoubtedly  be  a  presumption  in  such  a  case 
that  the  corner  was  correctly  fixed  by  the  government  surveyor 
where  the  field  notes  indicated  it  to  be. 


APPENDIX.  401 

2.  But  this  is  only  a  presumption,  and  may  be  overcome  by 
any  satisfactory  evidence  showing  that  in  fact  it  was  placed 
elsewhere. 

3.  No  statute  can  confer  upon  a  county  surveyor  the  power 
to  "establish"  corners,  and  thereby  bind  the  parties  concerned. 
Nor  is  this  a  question  merely  of  conflict  between  State  and 
Federal  law ;  it  is  a  question  of  property  right.     The  original 
surveys  must  govern,  and  the  laws  under  which  they  were  made 
must  govern,  because  the  land  was  bought  in  reference  to  them  ; 
and  any  legislation,  whether  State  or  Federal,  that  should  have 
the  effect  to  change  these,  would  be  inoperative,  because  dis- 
turbing vested  rights. 

4.  In  any  case  of  disputed  lines,  unless  the  parties  concerned 
settle  the  controversy  by  agreement,  the  determination  of  it  is 
necessarily  a  judicial  act,  and  it  must  proceed  upon  evidence, 
and  give  full  opportunity  for  a  hearing.     No  arbitrary  rules  of 
survey  or  of  evidence  can  be  laid  down  whereby  it  can  be  ad- 
judged.    The  general  duty  of  a  surveyor  in  such  a  case  is  plain 
enough.     He  is  not  to  assume  that  a  monument  is  lost,  until 
after  he  has  thoroughly  sifted  the  evidence,  and  found  himself 
unable  to  trace  it.     Even  then  he  should  hesitate  long  before 
doing  anything  to  the  disturbance  of  settled  possessions.    Occu- 
pation, especially  if  long  continued,  often  affords  very  satis- 
factory evidence  of  the  original  boundary,  when  no  other  is 
attainable  ;  and  the  surveyor  should  inquire  when  it  originated, 
how  and  why  the  lines  were  then   located  as  they  were,  and 
whether  a  claim  of  title  has  always  accompanied  the  possession, 
and  give  all  the  facts  due  force  as  evidence.    Unfortunately,  it  is 
known  that  surveyors  sometimes,  in  supposed  obedience  to  the 
State  statute,  disregard  all  evidences  of  occupation  and  claim 
of  title,  and  plunge  whole  neighborhoods  into  quarrels  and  liti- 
gation by  assuming  to  "establish"  corners  at  points  with  which 
the  previous  occupation  cannot  harmonize.     It  is  often  the  case 
that  where  one  or  more  corners  are  found  to  be  extinct,  all  par- 
ties concerned  have  acquiesced  in  lines  which  were  traced  by 
the  guidance  of  some  other  corner  or  landmark,  which  may  or 


402  PLANE   SURVEYING. 

may  not  have  been  trustworthy ;  but  to  bring  these  lines  into 
discredit,  when  the  people  concerned  do  not  question  them,  not 
only  breeds  trouble  in  the  neighborhood,  but  it  must  often  sub- 
ject the  surveyor  himself  to  annoyance,  and  perhaps  discredit, 
since  in  a  legal  controversy  the  law,  as  well  as  common  sense, 
must  declare  that  a  supposed  boundary  line  long  acquiesced  in 
is  better  evidence  of  where  the  real  line  should  be  than  any 
survey  made  after  the  original  monuments  have  disappeared. 
Stewart  v.  Carleton,  31  Mich.  Reports,  270;  Diehl  v.  Zanger, 
39  Mich.  Reports,  601.  And  county  surveyors,  no  more  than 
any  others,  can  conclude  parties  by  their  surveys. 

The  mischiefs  of  overlooking  the  facts  of  possession  must 
often  appear  in  cities  and  villages.  In  towns  the  block  and  lot 
stakes  soon  disappear ;  there  are  no  witness  trees  and  no  monu- 
ments to  govern,  except  such  as  have  been  put  in  their  places, 
or  where  their  places  were  supposed  to  be.  The  streets  are 
likely  to  be  soon  marked  off  b}*  fences,  and  the  lots  in  a  block 
will  be  measured  off  from  these  without  looking  farther. 

Now  it  may  perhaps  be  known  in  a  particular  case  that  a 
certain  monument  still  remaining  was  the  starting-point  in  the 
original  survey  of  the  town  plat ;  or  a  surveyor  settling  in  the 
town  may  take  some  central  point  as  the  point  of  departure  in 
his  surveys,  and  assuming  the  original  plat  to  be  accurate,  he  will 
then  undertake  to  find  all  streets  and  all  lots  by  course  and  dis- 
tance according  to  the  plat,  measuring  and  estimating  from  his 
point  of  departure.  This  procedure  might  unsettle  every  line 
and  every  monument  existing  by  acquiescence  in  the  town  ;  it 
would  be  very  likely  to  change  the  lines  of  streets,  and  raise 
controversies  everywhere.  Yet  this  is  what  is  sometimes 
done ;  the  surveyor  himself  being  the  first  person  to  raise  the 
disturbing  questions. 

Suppose,  for  example,  a  particular  village  street  has  been 
located  by  acquiescence  and  used  for  many  years,  and  the  pro- 
prietors in  a  certain  block  have  laid  off  their  lots  in  reference 
to  this  practical  location.  Two  lot-owners  quarrel,  and  one  of 
them  calls  in  a  surveyor  that  he  may  be  sure  that  his  neighbor 


APPENDIX.  403 

shall  not  get  an  inch  of  land  from  him.  This  surveyor  under- 
takes to  make  his  survey  accurate,  whether  the  original  was  or 
not,  and  the  first  result  is,  he  notifies  the  lot-owners  that  there 
is  error  in  the  street  line,  and  that  all  fences  should  be  moved, 
say,  one  foot  to  the  east.  Perhaps  he  goes  on  to  drive  stakes 
through  the  block  according  to  this  conclusion.  Of  course  if 
he  is  right  in  doing  this,  all  lines  in  the  village  will  be  unsettled  ; 
but  we  will  limit  our  attention  to  the  single  block.  It  is  not 
likely  that  the  lot-owners  will  generally  allow  the  new  survey  to 
unsettle  their  possessions,  but  there  is  always  a  probability  of 
finding  some  one  disposed  to  do  so.  We  shall  then  have  a  law- 
suit; and  with  what  result?  It  is  a  common  error  that  lines 
do  not  become  fixed  by  acquiescence  in  a  less  time  than  twenty 
years.  In  fact,  by  statute  road  lines  may  become  conclusively 
fixed  in  ten  years  ;  and  there  is  no  particular  time  that  shall  be 
required  to  conclude  private  owners,  where  it  appears  that  they 
have  accepted  a  particular  line  as  their  boundary,  and  all  con- 
cerned have  cultivated  and  claimed  up  to  it.  McNamara  v. 
Seaton,  82  111.  Reports,  498 ;  Bunce  v.  Bidwell,  43  Mich.  Re- 
ports, 542.  Public  policy  requires  that  such  lines  be  not  lightly 
disturbed  or  disturbed  at  all  after  the  lapse  of  any  considerable 
time.  The  litigant,  therefore,  who  in  such  a  case  pins  his 
faith  on  the  surveyor,  is  likely  to  suffer  for  his  reliance,  and 
the  surveyor  himself  to  be  mortified  by  a  result  that  seems  to 
impeach  his  judgment. 

Of  course  nothing  in  what  has  been  said  can  require  a  sur- 
veyor to  conceal  his  own  judgment  or  to  report  the  facts  one 
way  when  he  believes  them  to  be  another.  He  has  no  right  to 
mislead,  and  he  may  rightfully  express  his  opinion  that  an  origi- 
nal monument  was  at  one  place,  when  at  the  same  time  he  is 
satisfied  that  acquiescence  has  fixed  the  rights  of  parties  as  if 
it  were  at  another.  But  he  would  do  mischief  if  he  were  to 
attempt  to  "  establish"  monuments  which  he  knew  would  tend 
to  disturb  settled  rights  ;  the  farthest  he  has  a  right  to  go  as 
an  officer  of  the  law  is  to  express  his  opinion  where  the  monu- 
ment should  be  at  the  same  time  that  he  imparts  the  information 


404  PLANE   SURVEYING. 

to  those  who  employ  him,  and  who  might  otherwise  be  misled, 
that  the  same  authority  that  makes  him  an  officer,  and  entrusts 
him  to  make  surveys,  also  allows  parties  to  settle  their  own 
boundary  lines,  and  considers  acquiescence  in  a  particular  line 
or  monument  for  any  considerable  period  as  strong,  if  not  con- 
clusive, evidence  of  such  settlement.  The  peace  of  the  com- 
munity absolutely  requires  this  rule.  Foyce  v.  Williams,  26 
Mich.  Reports,  332.  It  is  not  long  since  that  in  one  of  the 
leading  cities  of  the  State  an  attempt  was  made  to  move  houses 
two  or  three  rods  into  a  street,  on  the  ground  that  a  survej', 
under  which  the  street  had  been  located  for  many  years,  had 
been  found  on  a  more  recent  survey  to  be  erroneous. 

From  the  foregoing  it  will  appear  that  the  duty  of  the  sur- 
veyor, where  boundaries  are  in  dispute,  must  be  varied  by  the 
circumstances.  (1)  He  is  to  search  for  original  monuments,  or 
for  the  places  where  they  were  originally  located,  and  allow 
these  to  control  if  he  finds  them,  unless  he  has  reason  to  believe 
that  agreements  of  the  parties  express  or  implied  have  rendered 
thtm  unimportant.  By  monuments  in  the  case  of  government 
surveys  we  mean,  of  course,  the  corner  and  quarter  stakes ; 
blazed  lines  or  marked  trees  on  the  lines  are  not  monuments  ; 
they  are  merely  guides  or  finger-posts,  if  we  may  use  the  ex- 
pression, to  inform  us  with  more  or  less  accuracy  where  the 
monuments  may  be  found.  (2)  If  the  original  monuments  are 
no  longer  discoverable,  the  question  of  location  becomes  one  of 
evidence  merely.  It  is  merely  idle  for  any  State  statute  to 
direct  a  surveyor  to  locate  or  ' '  establish "  a  corner,  as  the 
place  of  the  original  monument,  according  to  some  inflexible 
rule.  The  surveyor,  on  the  other  hand,  must  inquire  into  all 
the  facts,  giving  due  prominence  to  the  acts  of  parties  con- 
cerned, and  always  keeping  in  mind,  first,  that  neither  his  opin- 
ion nor  his  survey  can  be  conclusive  upon  parties  concerned  ; 
and,  second,  that  courts  and  juries  may  be  required  to  follow 
after  the  surveyor  over  the  same  ground,  and  that  it  is  exceed- 
ingly desirable  that  he  govern  his  action  by  the  same  lights  and 
same  rules  that  will  govern  theirs.  On  town  plats  if  a  surplus  or 


APPENDIX.  405 

deficiency  appears  in  a  block  when  the  actual  boundaries  are 
compared  with  the  original  figures,  and  there  is  no  evidence  to 
fix  the  exact  location  of  the  stakes  which  marked  the  division 
into  lots,  the  rule  of  common  sense  and  the  law  is  that  the  sur- 
plus or  deficiency  is  to  be  apportioned  between  the  lots  on  an 
assumption  that  the  error  extended  alike  to  all  parts  of  the 
block.  O'Brien  v.  McGrane,  29  Wis.  Reports,  446  ;  Quinnin  v. 
Reixers,  46  Mich.  Reports,  605. 

It  is  always  possible  when  corners  are  extinct  that  the  sur- 
veyor may  usefully  act  as  a  mediator  between  parties,  and  assist 
in  preventing  legal  controversies  by  settling  doubtful  Hues. 
Unless  he  is  made  for  this  purpose  an  arbitrator  by  legal  sub- 
mission, the  parties,  of  course,  even  if  they  consent  to  follow 
his  judgment,  cannot,  on  the  basis  of  mere  consent,  be  com- 
pelled to  do  so  ;  but  if  he  brings  about  an  agreement,  and  they 
carry  it  into  effect  by  actually  conforming  their  occupation  to 
his  lines,  the  action  will  conclude  them.  Of  course  it  is  desir- 
able that  all  such  agreements  be  reduced  to  writing ;  but  this  is 
not  absolutely  indispensable  if  they  are  carried  into  effect  with- 
out. 

Meander  Lines.  The  subject  to  which  allusion  will  now  be 
made  is  taken  up  with  some  reluctance,  because  it  is  believed 
the  general  rules  are  familiar.  Nevertheless,  it  is  often  found 
that  surveyors  misapprehend  them,  or  err  in  their  application  ; 
and  as  other  interesting  topics  are  somewhat  connected  with 
this,  a  little  time  devoted  to  it  will  probably  not  be  altogether 
lost.  The  subject  is  that  of  meander  lines.  These  are  lines 
traced  along  the  shores  of  lakes,  ponds,  and  considerable  rivers 
as  the  measures  of  quantity  when  sections  are  made  fractional 
by  such  waters.  These  have  determined  the  price  to  be  paid 
when  government  lands  were  bought,  and  perhaps  the  impres- 
sion still  lingers  in  some  minds  that  meander  lines  are  boundary 
lines,  and  all  in  front  of  them  remains  unsold.  Of  course  this 
is  erroneous.  There  was  never  any  doubt  that,  except  on  the 
large  navigable  rivers,  the  boundary  of  the  owners  of  the  banks 
is  the  middle  line  of  the  river  ;  and  while  some  courts  have  held 


406  PLANK  SURVEYING. 

that  this  was  the  rule  on  all  fresh-water  streams,  large  and  small, 
others  have  held  to  the  doctrine  that  the  title  to  the  bed  of  the 
stream  below  low-water  mark  is  in  the  State  while  conceding  to 
the  owners  of  the  bank  all  riparian  rights.  The  practical  differ- 
ence is  not  very  important.  In  this  State  the  rule  that  the  centre 
line  is  the  boundary  line  is  applied  to  all  our  great  rivers,  includ- 
ing the  Detroit,  varied  somewhat  by  the  circumstance  of  there 
being  a  distinct  channel  for  navigation  in  some  cases  with  the 
stream  in  the  main  shallow,  and  also  sometimes  by  the  existence 
of  islands. 

The  troublesome  questions  for  surveyors  present  themselves 
when  the  boundary  line  between  two  contiguous  estates  is  to  be 
continued  from  the  meander  line  to  the  centre  line  of  the  river. 
Of  course  the  original  survey  supposes  that  each  purchaser  of 
land  on  the  stream  has  a  water-front  of  the  length  shown  by  the 
field  notes  ;  and  it  is  presumable  that  he  bought  this  particular 
land  because  of  that  fact.  In  many  cases  it  now  happens  that 
the  meander  line  is  left  some  distance  from  the  shore  by  the 
gradual  change  of  course  of  the  stream  or  diminution  of  the  flow 
of  water.  Now  the  dividing  line  between  two  government  sub- 
divisions might  strike  the  meander  line  at  right  angles,  or  ob- 
liquely ;  and  in  some  cases,  if  it  were  continued  in  the  same 
direction  to  the  centre  line  of  the  river,  might  cut  off  from  tbe 
water  one  of  the  subdivisions  entirely,  or  at  least  cut  it  off  from 
any  privilege  of  navigation  or  other  valuable  use  of  the  water, 
while  the  other  might  have  a  water-front  much  greater  than  the 
length  of  a  line  crossing  it  at  right  angles  to  its  side  lines.  The 
effect  might  be  that,  of  two  government  subdivisions  of  equal 
size  and  cost,  one  would  be  of  very  great  value  as  water-front 
property,  and  the  other  comparatively  valueless.  A  rule  which 
would  produce  this  result  would  not  be  just,  and  it  has  not  been 
recognized  in  the  law. 

Nevertheless,  it  is  not  easy  to  determine  what  ought  to  be  the 
correct  rule  for  every  case.  If  the  river  has  a  straight  course, 
or  one  nearly  so,  every  man's  equities  will  be  preserved  by  this 
rule.  Extend  the  line  of  division  between  the  two  parcels  from 


APPENDIX.  407 

the  meander  line  to  the  centre  line  of  the  river,  as  nearly  as  pos- 
sible at  right  angles  to  the  general  course  of  the  river  at  that 
point.  This  will  preserve  to  each  man  the  water-front  which 
the  field  notes  indicated,  except  as  changes  in  the  water  may 
have  affected  it,  and  the  only  inconvenience  will  be  that  the 
division  line  between  different  subdivisions  is  likely  to  be  more 
or  less  deflected  where  it  strikes  the  meander  line. 

This  is  the  legal  rule,  and  it  is  not  limited  to  government  sur- 
veys, but  applies  as  well  to  water-lots  which  appear  as  such  on 
town  plats.  Bay  City  Gas  Light  Co.  v.  The  Industrial  Works, 
28  Mich.  Reports,  182.  It  often  happens,  therefore,  that  the 
lines  of  city  lots  bounded  on  navigable  streams  are  deflected  as 
they  strike  the  bank,  or  the  line  where  the  bank  was  when  the 
town  was  first  laid  out.  When  the  stream  is  very  crooked,  and 
especially  if  there  are  short  bends,  so  that  the  foregoing  rule  is 
incapable  of  strict  application,  it  is  sometimes  very  difficult  to 
determine  what  shall  be  done ;  and  in  many  cases  the  surveyor 
may  be  under  the  necessity  of  working  out  a  rule  for  himself. 
Of  course  his  action  cannot  be  conclusive  ;  but  if  he  adopts  one 
that  follows,  as  nearly  as  the  circumstances  will  admit,  the  gen- 
eral rule  above  indicated,  so  as  to  divide  as  near  as  may  be  the 
bed  of  the  stream  among  the  adjoining  owners  in  proportion  to 
their  lines  upon  the  shore,  his  division,  being  that  of  an  expert, 
made  upon  the  ground  and  with  all  available  lights,  is  likely  to 
be  adopted  as  law  for  the  case.  Judicial  decisions,  into  which 
the  surveyor  would  find  it  prudent  to  look  under  such  circum- 
stances, will  throw  light  upon  his  duties,  and  may  constitute  a 
sufficient  guide  when  peculiar  cases  arise.  Each  riparian  lot- 
owner  ought  to  have  a  line  on  the  legal  boundary,  namely,  the 
centre  line  of  the  stream,  proportioned  to  the  length  of  his  line 
on  the  shore  ;  and  the  problem  in  each  case  is,  how  this  is  to  be 
given  him.  Alluvion,  when  a  river  imperceptibly  changes  its 
course,  will  be  apportioned  by  the  same  rules. 

The  existence  of  islands  in  a  stream,  when  the  middle  line 
constitutes  a  boundary,  will  not  affect  the  apportionment  unless 
the  islands  were  surveyed  out  as  government  subdivisions  iu  the 


408  PLANE   SURVEYING. 

original  admeasurement.  Wherever  that  was  the  case  the  pur- 
chaser of  the  island  divides  the  bed  of  the  stream  on  each  side 
with  the  owner  of  the  bank,  and  his  rights  also  extend  above 
and  below  the  solid  ground,  and  are  limited  by  the  peculiarities 
of  the  bed  and  the  channel.  If  an  island  was  not  surveyed  as  a 
government  subdivision  previous  to  the  sale  of  the  bank,  it  is  of 
course  impossible  to  do  this  for  the  purposes  of  government  sale 
afterwards,  for  the  reason  that  the  rights  of  the  bank  owners  are 
fixed  by  their  purchase  :  when  making  that  they  have  a  right  to 
understand  that  all  land  between  the  meander  lines,  not  sepa- 
rately surveyed  and  sold,  will  pass  with  the  shore  in  the  govern- 
ment sale  ;  and  having  this  right,  anything  which  their  purchase 
would  include  under  it  cannot  afterwards  be  taken  from  them. 
It  is  believed,  however,  that  the  Federal  courts  would  not  recog- 
nize the  applicability  of  this  rule  to  large  navigable  rivers,  such 
as  those  uniting  the  Great  Lakes. 

On  all  the  little  lakes  of  the  State,  which  are  mere  expansions 
near  their  mouths  of  the  rivers  passing  through  them,  —  such  as 
the  Muskegon,  Pere  Marquette,  and  Manistee,  7—  the  same  rule  of 
bed  ownership  has  been  judicially  applied  that  is  applied  to  the 
rivers  themselves ;  and  the  division  lines  are  extended  under 
the  water  in  the  same  way.  Rice  v.  Ruddiman,  10  Mich.  125. 
If  such  a  lake  were  circular,  the  lines  would  converge  to  the 
centre  ;  if  oblong  or  irregular,  there  might  be  a  line  in  the  middle 
on  which  they  would  terminate,  whose  course  would  bear  some 
relation  to  that  of  the  shore.  But  it  can  seldom  be  important 
to  follow  the  division  line  very  far  under  the  water,  since  all 
private  rights  are  subject  to  the  public  rights  of  navigation  and 
other  use,  and  any  private  use  of  the  lands  inconsistent  with 
these  would  be  a  nuisance,  and  punishable  as  such.  It  is  some- 
times important,  however,  to  run  the  lines  out  for  some  consid- 
erable distance,  in  order  to  determine  where  one  may  lawfully 
moor  vessels  or  rafts  for  the  winter,  or  cut  ice.  The  ice  crop 
that  forms  over  a  man's  land  of  course  belongs  to  him.  Lormau 
v.  Benson,  8  Mich.  18 ;  People's  Ice  Co.  v.  Steamer  Excelsior, 
recently  decided. 


APPENDIX.  409 

What  is  said  above  will  show  how  unfounded  is  the  notion, 
which  is  sometimes  advanced,  that  a  ripai-ian  proprietor  on  a 
meandered  river  may  lawfully  raise  the  water  in  the  stream 
without  liability  to  the  proprietors  above,  provided  he  does  not 
raise  it  so  that  it  overflows  the  meander  line.  The  real  fact  is, 
that  the  meander  line  has  nothing  to  do  with  such  a  case,  and 
an  action  will  lie  whenever  he  sets  back  the  water  upon  the 
proprietor  above,  whether  the  overflow  be  below  the  meander 
lines  or  above  them.  As  regards  the  lakes  and  ponds  of  the 
State,  one  may  easily  raise  questions  that  it  would  be  impossible 
for  him  to  settle.  Let  us  suggest  a  few  questions,  some  of 
which  are  easily  answered,  and  some  not :  (1)  To  whom  belongs 
the  land  under  these  bodies  of  water,  where  they  are  not  mere 
expansions  of  a  stream  flowing  through  them?  (2)  What 
public  rights  exist  in  them?  (3)  If  there  are  islands  in  them 
which  were  not  surveyed  out  and  sold  by  the  United  States,  can 
this  be  done  now?  Others  will  be  suggested  by  the  answers 
given  to  these. 

It  seems  obvious  that  the  rules  of  private  ownership  which 
are  applied  to  rivers  cannot  be  applied  to  the  Great  Lakes. 
Perhaps  it  should  be  held  that  the  boundary  is  at  low-water 
mark,  but  improvements  beyond  this  would  only  become  unlaw- 
ful when  they  became  nuisances.  Islands  in  the  Great  Lakes 
would  belong  to  the  United  States  until  sold,  and  might  be 
surveyed  and  measured  at  any  time.  The  right  to  take  fish  in 
the  lakes  or  to- cut  ice  is  public,  like  the  right  of  navigation, 
but  is  to  be  exercised  in  such  manner  as  not  to  interfere  with 
the  rights  of  shore-owners ;  but,  so  far  as  these  public  rights 
can  be  the  subject  of  ownership,  they  belong  to  the  State,  not 
the  United  States  ;  and  so,  it  is  believed,  does  the  bed  of  a  lake 
also.  Pollard  v.  Hagan,  3  Howard's  U.  S.  Reports.  But 
such  rights  are  not  generally  considered  proper  subjects  of 
sale,  but,  like  the  right  to  make  use  of  the  public  highways, 
they  are  held  by  the  State  in  trust  for  all  the  people.  What  is 
said  of  the  large  lakes  may,  perhaps,  be  said  also  of  many  of  the 
interior  lakes  of  the  State ;  such,  for  example,  as  Houghton, 


410  PLANE   SURVEYING. 

Higgins,  Cheboygan,  Burt's,  Mullet,  Whitmore,  and  many 
others.  But  there  are  many  little  lakes  or  ponds  which  are 
gradually  disappearing,  and  the  shore  proprietorship  advances 
pari  passu  as  the  waters  recede.  If  these  are  of  any  consider- 
able size,  —  say,  even  a  mile  across,  —  there  may  be  questions 
of  conflicting  rights  which  no  adjudication  hitherto  made  could 
settle.  Let  an}'  surveyor,  for  example,  take  the  case  of  a  pond 
of  irregular  form,  occupying  a  mile  square  or  more  of  territory, 
and  undertake  to  determine  the  rights  of  the  shore  proprietors 
to  its  bed  when  it  shall  totally  disappear,  and  he  will  find  lie  is 
in  the  midst  of  problems  such  as  probably  he  has  never  grappled 
with,  or  reflected  upon,  before.  But  the  general  rules  for  the 
extension  of  shore  lines  which  have  already  been  laid  down 
should  govern  such  cases,  or  at  least  should  serve  as  guides  in 
their  settlement. 

Where  a  pond  is  so  small  as  to  be  included  within  the  lines 
of  a  private  purchase  from  the  government,  it  is  not  believed 
the  public  have  any  rights  in  it  whatever.  Where  it  is  not  so 
included,  it  is  believed  they  have  rights  of  fishery,  rights  to 
take  ice  and  water,  and  rights  of  navigation  for  business  or 
pleasure.  This  is  the  common  belief,  and  probably  the  just 
one.  Shore  rights  must  not  be  so  exercised  as  to  disturb  these, 
and  the  States  may  pass  all  proper  laws  for  their  protection. 
It  would  be  easy  with  suitable  legislation  to  preserve  these  little 
bodies  of  water  as  permanent  places  of  resort  for  the  pleasure 
and  recreation  of  the  people,  and  there  ought  to  be  such  legisla- 
tion. If  the  State  should  be  recognized  as  owner  of  the  beds 
of  these  small  lakes  and  ponds,  it  would  not  be  owner  for  the 
purpose  of  selling.  It  would  be  owner  only  as  a  trustee  for  the 
public  use;  and  a  sale  would  be  inconsistent  with  the  right  of 
the  bank  owners  to  make  use  of  the  water  in  its  natural  condi- 
tion in  connection  with  their  estates.  Some  of  them  might  be 
made  salable  lands  by  draining ;  but  the  State  could  not  drain, 
even  for  this  purpose,  against  the  will  of  the  shore-owners,  un- 
less their  rights  were  appropriated  and  paid  for.  Upon  many 
questions  that  might  arise  between  the  State  as  owner  of  the 


APPENDIX.  4H 

bed  of  a  little  lake  and  the  shore-owners,  it  would  be  presump- 
tuous to  express  an  opinion  now,  and  fortunately  the  occasion 
does  not  require  it. 

I  have  thus  indicated  a  few  of  the  questions  with  which  sur- 
veyors may  now  and  then  have  occasion  to  deal,  and  to  which 
they  should  bring  good  sense  and  ssund  judgment.  Surveyors 
are  not,  and  cannot  be,  judicial  officers,  but  in  a  great  many 
cases  they  act  in  a  quasi  judicial  capacity,  with  the  acquies- 
cence of  parties  concerned;  and  it  is  important  for  them  to 
know  by  what  rules  they  are  to  be  guided  in  the  discharge  of 
their  judicial  functions.  What  I  have  said  cannot  contribute 
much  to  their  enlightenment,  but  I  trust  will  not  be  wholly 
without  value. 


TABLES. 


TABLE  I. 

THE 

COMMON  OR  BRIGGS  LOGARITHMS 

OF    THE 

NATURAL   NUMBERS 

From  1  to  10000. 

1-100 

N 

log 

I 

log 

K 

log 

N 

log 

* 

log 

1 

0.00000 

21 

1.  32  222 

41 

1.61278 

61 

1.78533 

81 

1.90849 

2 

0.  30  103 

22 

1.  34  242 

42 

1.  62  325 

62 

1.  79  239 

82 

1.91381 

3 

0.47712 

23 

1.  36  173 

43 

1.  63  347 

63 

1.79934 

83 

1.91908 

4 

0.60206 

24 

1.38021 

44 

1.  64  345 

64 

1.80618 

84 

1.92428 

5 

0.  69  897 

25 

1.  39  794 

45 

1.  65  321 

65 

1.  81  291 

85 

1.92942 

6 

0.  77  815 

26 

1.41497 

46 

1.  66  276 

66 

1.  81  954 

86 

1.93450 

7 

0.  84  510 

27 

1.43136 

47 

1.  67  210 

67 

1.82607 

87 

1.93952 

8 

0.90309 

28 

1.44716 

48 

1.  68  124 

68 

1.  83  251 

88 

1.94448 

9 

0.  95  424 

29 

1.46240 

49 

1.69020 

69 

1.83885 

89 

1.94939 

10 

1.00000 

30 

1.  47  712 

50 

1.69897 

70 

1.84510 

90 

1.95424 

11 

1.  04  139 

31 

1.49136 

51 

1.  70  757 

71 

1.  85  126 

91 

1.95904 

12 

1.07918 

32 

1.50515 

52 

1.71600 

72 

1.  85  733 

92 

1.96379 

13 

1.  11  394 

33 

1.51851 

63 

1.  72  428 

73 

1.86332 

93 

1.96848 

14 

1.14613 

34 

1.  53  148 

54 

1.  73  239 

74 

1.86923 

94 

1.97313 

15 

1.17609 

35 

1.54407 

55 

1.74036 

75 

1.  87  506 

95 

1.  97  772 

16 

1.20412 

36 

1.  55  630 

56 

1.74819 

76 

1.88081 

96 

1.98227 

17 

1.  23  045 

37 

1.56820 

67 

1.  75  587 

77 

1.88649 

97 

1.98677 

18 

1.  25  527 

38 

1.57978 

58 

1.76343 

78 

1.89209 

98 

1.99123 

19 

1.  27  875 

39 

1.59106 

59 

1.  77  085 

79 

1.89763 

99 

1.99564 

20 

1.  30  103 

40 

1.60206 

60 

1.77815 

80 

1.90309 

100 

2.00000 

I 

log 

« 

log 

I 

log 

V 

log 

N 

N 

1-100 


100-150 


If 

01234 

56789 

100 

101 
102 
103 
104 

00000  00043  00087  00130  00173 
00432  00475  00518  00561   00604 
00860  00903  00945  00988  01030 
01284  01326  01368  01410  01452 
01703  01745  01787  01828  01870 

00217  00260  00303  00346  00389 
00647  00689  00732  00775  00817 
01  072  01  115  01  157  01  199  01  242 
01494  01536  01578  01620  01662 
01912  01953  01995  02036  02078 

106 
106 
107 
108 
109 

02119  02160  02202  02243  02284 
02531  02572  02612  02653  02694 
02938  02979  03019  03060  03100 
03342  03383  03423  03463  03503 
03743  03782  03822  03862  03902 

02325  02366  02407  02449  02490 
02735  02776  02816  02857  02898 
03141  03181  03222  03262  03302 
03543  03583  03623  03663  03703 
03941  03981  04021  04060  04100 

110 

111 
112 
113 

114 

04139  04179  04218  04258  04297 
04532  04571  04610  04650  04689 
04922  04961  04999  05038  05077 
05  308  05  346  05  385  05  423  05  461 
05690  05729  05767  05805  05843 

04336  04376  04415  04454  04493 
04727  04766  04805   04844  04883 
05115  05154  05192  05231   05269 
05500  05538  05576  05614  05652 
05881  05918  05956  05994  06032 

115 
116 

117 
118 
119 

06070  06108  06145  06183  06221 
06446  06483  06521  06558  06595 
06819  06856  06893  06930  06967 
07188  07225  07262  07298  07335 
07555  07591  07628  07664  07700 

06258  06296  06333  06371  06408 
06633  06670  06707  06744  06781 
07004  07041   07078  07115  07151 
07372  07408  07445  07482  07518 
07737  07773  07809  07846  07882 

120 

121 
122 
123 

124 

07918  07954  07990  08027  08063 
08279  08314  08350  08386  08422 
08636  08672  08707  08743  08778 
08991  09026  09061   09096  09132 
09342  09377  09412  09447  09482 

08099  08135  08171   08207  08243 
08458  08493  08529  08565  08600 
08814  08849  08884  08920  08955 
09167  09202  09237  09272  09307 
09517  09552  09587  09621  09656 

125 
126 
127 
128 
129 

09691  09726  09760  09795  09830 
10037  10072  10106  10140   10175 
10380  10415   10449  10483   10517 
10721   10755  10789  10823   10857 
11059  11093  11126  11160  11193 

09864  09899  09934  09968  10003 
10209   10243   10278  10312   10346 
10551   10585   10619  10653   10687 
10890  10924   10958  10992   11025 
11227  11261   11294  11327   11361 

ISO 

131 
132 
133 
134 

11394  11428  11461   11494   11528 
11727  11760  11793   11826  11860 
12057  12090  12123   12156   12189 
12385   12418  12450  12483   12516 
12710  12743  12775  12808  12840 

11561   11594  11628   11661    11694 
11893   11926  11959  11992   12024 
12222   12254   12287  12320   12352 
12548  12581   12613   12646  12678 
12872  12905   12937  12969  13001 

135 
136 
137 
138 
139 

13033  13066  13098  13130  13162 
13354  13386  13418  13450   13  481 
13672   13704  13735   13767   13799 
13988  14019  14051   14082   14114 
14301   14333   14364  14395    14426 

13194  13226  13258  13290  13322 
13513   13545   13577  13609  13640 
13830  13862   13893   13925   13956 
14145    14176   14208   14239  14270 
14457  14489  14520  14551   14582 

14O 

141 
142 
143 
144 

14613  14644  14675   14706   14737 
14922  14953   14983   15014  15045 
15229  15259  15290   15320  15351 
15534  15564  15594   15625   15655 
15836  15866  15897   15927  15957 

14768  14799  14829  14860  14891 
15076  15106  15137  15168  15198 
15381    15412   15442  15473   15503 
15685   15715   15746  15776  15806  ' 
15  987   16017   16047  16077  16107 

145 
146 
147 
148 
149 

16137  16167  16197   16227   16256 
16435   16465   16495   16524   16554 
16732  16761   16791   16820  16850 
17026  17056  17085   17114   17143 
17319  17348  17377  17406  17435 

16286  16316   16346  16376  16406 
16584   16613   16643   16673   16702 
16879  16909   16938  16967  16997 
17173   17202   17231   17260  17289 
17464  17493   17522   17551   17580 

ISO 

17609  17638  17667  17696  17725 

17754   17782   17811   17840  17S69 

N 

01234 

36789 

100-150 


150-200 


N 

O           1           2           3           4 

56789 

ISO 

17609  17638   17667   17696   17725 

17754   17782   17811   17840  17869 

151 

17898  17926  17955   17984   18013 

18041   18070  18099  18127  18156 

152 

18  184  18213   18241   18270  18298 

18327  18355   18384  18412   18441 

153 

18469  18498   18526  18554  18583 

18611   18639  18667  186%  18724 

154 

18752  18780  18808  18837  18  865. 

18893   18921   18949  18977   19005 

155 

19033   19061   19089  19117  19145 

19173   19201   19229  19257   19285 

156 

19312   19340  19368   19396  19424 

19451   19479  19507   19535   19562 

157 

19590  19618   19645    19673   19700 

19728  19756  19783   19811  19838 

158 

19866   19893   19921   19948  19976 

20003  20030  20058  20085  20112 

159 

20140  20167  20194  20222  20249 

20276  20303  20330  20358  20385 

160 

20412  20439  20466  20493  20520 

20548  20575  20602  20629  20656 

161 

20683  20710  20737  20763  20790 

20817  20844  20871   20898  20925 

162 

20952  20978  21005  21032  21059 

21085  21112  21139  21165  21192 

163 

21219  21245  21272  21299  21325 

21352  21378  21405  21431   21458 

164 

21484  21511  21537  21564  21590 

21617  21643  21669  216%  21722 

165 

21748  21775  21801   21827  21854 

21880  21906  21932  21958  21985 

166 

22011   22037  22063   22089  22115 

22141   22167   22194  22220  22246 

167 

22272  22298  22324  22350  22376 

22401  22427  22453  22479  22505 

168 

22531  22557  22583   22608  22634 

22660  22686  22712  22737  22763 

169 

22789  22814  22840  22866  22891 

22917  22943  22968  22994  23019 

17O 

23045  23070  23096  23121  23147 

23172  23198  23223  23249  23274 

171 

23300  23325   23350  23376  23401 

23426  23452  23477  23502  23528 

172 

23553  23578  23603   23629  23654 

23679  23704  23729  23754  23779 

173 

23805   23830  23855   23880  23905 

23930  23955   23980  24005  24030 

174 

24055  24080  24105   24130  24155 

24180  24204  24229  24254  24279 

175 

24304  24329  24353  24378  24403 

24428  24452  24477  24502  24527 

176 

24551  24576  24601   24625   24650 

24674  24699  24724  24748  24773 

177 

24797  24822  24846  24871   24895 

24920  24944  24969  24993  25018 

178 

25042  25066  25091   25115  25139 

25164  25188  25212  25237  25261 

179 

25285  25310  25334  25358  25382 

25406  25431  25455  25479  25503 

180 

25527  25551  25575  25600  25624 

25648  25672  25696  25720  25744 

181 

25768  25792  25816  25840  25864 

25888  25912  25935   25959  25983 

182 

26007  26031   26055   26079  26102 

26126  2615TT26174  26198  26221 

183 

26245   26269  26293   26316  26340 

26364  26387  26411   26435  26458 

184 

26482  26505  26529  26553   26576 

26600  26623  26647  26670  26694 

185 

26717  26741  26764  26788  26811 

26834  26858  26881  26905  26928 

186 

26951   26975  26998  27021  27045 

27068  27091  27114  27138  27161 

187 

27184  27207  27231   27254  27277 

27300  27323  27346  27370  27393 

188 

27416  27439  27462  27485   27508 

27531  27554  27577  27600  27623 

189 

27646  27669  27692  27715  27738 

27761  27784  27807  27830  27852 

190 

27875  27898  27921  27944  27  967 

27989  28012  28035  28058  28081 

191 

28103   28126  28149  28171   28194 

28217  28240  28262  28285  28307 

192 

28330  28353  28375   28398  28421 

28443  28466  28488  28511   28533 

193 

28556  28578  28601   28623   28646 

28668  28691   28713  28735   28758 

194 

28780  28803   28825   28847  28870 

28892  28914  28937  28959  28  981 

195 

29003   29026  29048  29070  29092 

29115  29137  29159  29181  29203 

196 

29226   29248  29270  29292   29314 

29336  29358  29380  29403  29425 

197 

29447  29469  29491   29513  29535 

29557  29579  29601  29623  29645 

198 

29667  29  688  29710  29732  29754 

29776  29798  29820  29842  29863 

199 

29885   29907  29929  29951   29973 

29994  30016  30038  30060  300S1 

200 

30103   30125   30146.30168   30190 

30211  30233  30255  30276  30298 

N 

O            1            2            3            4 

5           6           7          8          i> 

150-200 


200-250 


N 

01234 

5          6           78          9 

200 
201 
202 
203 
204 

30103  30125  30146  30168  30190 
30320  30341  30363  30384  30406 
30535  30557  30578  30600  30621 
30750  30771  30792  30814  30835 
30963  30984  31006  31027  31048 

30211  30233  30255   30276  30298 
30428  30449  30471  30492  30514 
30643  30664  30685  30707  30728 
30856  30878  30899  30920  30942 
31069  31091  31112  31133  31154 

205 
206 
207 
208 
209 

31175  31197  31218  31239  31260 
31387  31408  31429  31450  31471 
31597  31618  31639  31660  31681 
31806  31827  31848  31  869  31890 
32015  32035  32056  32077  32098 

31281  31302  31323   31345  31366 
31492  31513  31534  31555  31576 
31702  31723  31744  31765  31785 
31911  31931  31952  31973  31994 
32118  32139  32160  32181  32201 

21O 

211 
212 
213 
214 

32222  32243  32263  32284  32305 
32428  32449  32469  32490  32510 
32634  32654  32675  32695  32715 
32838  32858  32879  32899  32919 
33041   33062  33082  33102  33122 

32325  32346  32366  32387  32408 
32531  32552  32572  32593  32613 
32  736  32  756  32  777  32  797  32  818 
32940  32960  32  980  33001   33021 
33143  33163  33183  33203  33224 

215 
216 
217 
218 
219 

33244  33264  33284  33304  33325 
33445  33465  33486  33506  33526 
33646  33666  33686  33706  33726 
33846  33866  33885  33905  33925 
34044  34064  34084  34104  34124 

33345  33365  33385  33405  33425 
33  546  33  566  33  586  33  606  33  626 
33746  33766  33786  33806  33826 
33945  33965   33985  34005   34025 
34143  34163  34183  34203  34223 

220 

221 
222 
223 
224 

34242  34262  34282  34301  34321 
34439  34459  34479  34498  34518 
34635  34655  34674  34694  34713 
34830  34850  34869  34889  34908 
35025  35044  35064  35083  35102 

34341  34361  34380  34400  34420 
34537  34557  34577  345%  34616 
34733  34753  34772  34792  34811 
34928  34947  34967  34986  35005 
35122  35141  35160  35180  35199 

226 
226 
227 
228 
229 

35218  35238  35257  35276  35295 
35411  35430  35449  35468  35488 
35603  35622  35641  35660  35679 
35793  35813  35832  35851  35870 
35984  36003  36021  36040  36059 

35315  35334  35353  35372  35392 
35507  35526  35545  35564  35583 
35  698  35  717  35  736  35  755  35  774 
35889  35908  35927  35946  35965 
36078  36097  36116  36135  36154 

230 

231 
232 
233 
234 

36173  36192  36211  36229  36248 
36361  36380  36399  36418  36436 
36549  36568  36586  36605  36624 
36736  36754  36773  36791  36810 
36922  36940  36959  36977  369% 

36267  36286  36305  36324  36342 
36455  36474  36493  36511  36530 
36642  36661   36680  36698  36717 
36829  36847  36866  36884  36903 
37014  37033  37051  37070  37088 

235 
236 
237 
238 

239 

37107  37125  37144  37162  37181 
37291  37310  37328  37346  37365 
37475  37493  37511  37530  37548 
37658  37676  37694  37712  37731 
37840  37858  37876  37894  37912 

37199  37218  37236  37254  37273 
37383  37401  37420  37438  37457 
37566  37585  37603  37621  37639 
37749  37767  37785  37803  37822 
37931  37  949-  37  %7  37985  38003 

240 

241 
242 
243 
244 

38021  38039  38057  38075  38093 
38202  38220  38238  38256  38274 
38382  38399  38417   38435  38453 
38561  38578  385%  38614  38632 
38739  38757  38775  38792  38810 

38112  38130  38148  38166  38184 
38292  38310  38328  38346  38364 
38471  38489  38507  38525  38543  ' 
38650  38668  38686  38703   38721 
38828  38846  38863  38881   38899    , 

245 
246 
247 
248 
249 

38917  38934  38952  38970  38987 
39094  39111  39129  39146  39164 
39270  39287  39305   39322  39340 
39445  39463  39  480  39498  39515 
39620  39637  39655  39672  39690 

39005  39023  39041  39058  39  076  j 
39182  39199  39217  39235    39252 
39358  39375  39393  39410  39428 
39533  39550  39568  39585   39602 
39  707  39  724  39  742  39  759  39  777 

250 

39794  39811  39829  39846  39863 

39881  39898  39915  39933  39950 

X 

01234 

56789 

200-250 


250-300 


0 


8 


25O 

251 
252 
253 
254 

255 
256 
257 
268 
259 

260 

261 
262 
263 

264 

265 


268 
269 

270 

271 
272 
273 
274 

275 
276 
277 
278 
279 


283 
284 

285 
286 
287 
288 
289 

29O 

291 
292 
293 
294 

295 
296 
297 
298 


39794  39811  39829 
39967  39985  40002 
40140  40157  40175 
40312  40329  40346 
40483  40500  40518 

40654  40671  40688 

40824  40841  40858 

40993  41010  41027 

41162  41179  41196 

41330  41347  41363 


39846  39863 
40019  40037 
40192  40209 
40364  40381 
40535  40552 

40705  40722 

40875  40892 

41044  41061 

41212  41229 

41380  41397 


41497  41514  41531  41547  41564 

41664  41681  41697  41714  41731 

41830  41847  41863  41880  418% 

41996  42012  42029  42045  42062 

42160  42177  42193  42210  42226 

42325  42341  42357  42374  42390 

42  488  42504  42521   42537  42553 

42651  4266V  42684  42700  42716 

42813  42830  42846  42862  42878 

42975  42991  43008  43024  43040 

43136  43152  43169  43185  43201 

43297  43313  43329  43345   43361 

43457  43473  43489  43505   43521 

43616  43632  43648  43664  43680 

43775  43791  43807  43823  43838 

43933  43949  43965  43981  439% 

44091  44107  44122  44138  44154 

44248  44264  44279  44295  44311 

44404  44420  44436  44451  44467 

44560  44576  44592  44607  44623 

44716  44731  44747  44762  44778 

44871  44886  44902  44917  44932 

45025  45040  45056  45071   45086 

45  179  45  194  45  209  45  225  45  240 

45332  45347  45362  45378  45393 

45484  45500  45515  45530  45545 

45637  45652  45667  45682  45697 

45788  45803  45818  45834  45849 

45939  45954  45969  45984  46000 

46090  46105  46120  46135  46150 

46240  46255  46270  46285  46300 

46389  46404  46419  46434  46449 

46538  46553  46568  46583  46598 

46687  46702  46716  46731  46746 

46835  46850  46864  46879  46894 

46982  46997  47012  47026  47041 

47129  47144  47159  47173  47188 

47276  47290  47305  47319  47334 

47422  47436  47451  47465  47480 

47567  47582  47596  47611  47625 

47712  47727  47741  47756  47770 


39881  39898  39915 

40054  40071  40088 

40226  40243  40261 

40398  40415  40432 

40569  40586  40603 

40739  40756  40773 
40909  40926  40943 
41 078  41 095  41  111 
41246  41263  41280 
41414  41430  41447 


39933  39950 
40106  40123 
40278  40295 
40449  40466 
40620  40637 

40790  40807 

40960  40976 

41128  41 14* 

412%  41313 

41464  41481 


41581  41597  41614  41631  41647 
41  747  41  764  41  780  41  797  41  814 
41913  41929  41946  41963  41979 
42078  42095  42111  42127  42144 
42243  42259  42275  42292  42308 

42406  42423  42439  42455  42472 

42570  42586  42602  42619  42635 

42732  42749  42765  42781  42797 

42894  42911  42927  42943   42959 

43056  43072  43088  43104  43120 

43  217  .43  233  43  249  43  265  43  281 
43377  43393  43409  43425  43441 
43537  43553  43569  43584  43600 
43696  43712  43727  43743  43759 
43854  43870  43886  43902  43917 

44012  44028  44044  44059  44075 

44170  44185  44201  44217  44232 

44326  44342  44358  44373  44389 

44483  44498  44514  44529  44545 

44638  44654  44669  44685  44700 

44793  44809  44824  44840  44855 

44948  44963  44979  44994  45010 

45  102  45  117  45  133  45  148  45  163 

45255  45271  45  2S6  45301   45317 

45408  45423  45439  45454  45469 

45  561  45  576  45  591  45  606  45  621 

45712  45728  45743  45758  45773 

45864  45879  45894  45909  45924 

46015  46030  46045  46060  46075 

46165  46180  46195  46210  46225 

46315  46330  46345  46359  46374 
46464  46479  46494  46509  46523 
46613  46627  46642  46657  46672 
46761  46776  46790  46805  46820 
46909  46923  46938  46953  46%7 

47056  47070  47085  47100  47114 
47202  47217  47232  47246  47261 
47349  47363  47378  47392  47407 
47494  47509  47524  47538  47553 
47640  47654  47669  47683  47698 

47784  47799  47813  47828  47842 


250-300 


300-350 


N 

01234 

56789 

300 

301 
302 
303 
304 

47712  47727  47741  47756  47770 
47857  47871  47885  47900  47914 
48001  48015  48029  48044  48058 
48144  48159  48173  48187  48202 
48287  48302  48316  48330  48344 

47784  47799  47813  47828  47842 
47929  47943  47958  47972  47986 
48073  48087  48101  48116  48130 
48216  48230  48244  48259  48273 
48359  48373  48387  48401  48416 

305 
306 
307 
308 
309 

48430  48444  48458  48473  48487 
48572  48586  48601   48615  48629 
48714  48728  48742  48756  48770 
48855  48869  48883  48897  48911 
48996  49010  49024  49038  49052 

48501  48515  48530  48544  48558 
48643  48657  48671    48686  48700 
48785  48799  48813  48827  48841 
48926  48940  48954  48968  48982 
49066  49080  49094  49108  49122 

310 

311 
312 
313 
314 

49136  49150  49164  49178  49192 
49276  49290  49304  49318  49332 
49415  49429  49443  49457  49471 
49554  49568  49582  49596  49610 
49693  49707  49721  49734  49748 

49206  49220  49234  49248  49262 
49346  49360  49374  49388  49402 
49485  49499  49513  49527  49541 
49624  49638  49651  49665  49679 
49762  49776  49790  49803  49817 

315 
316 
317 
318 

319 

49831  49845   49859  49872  49886 
49969  49982  49996  50010  50024 
50106  50120  50133  50147  50161 
50243  50256  50270  50284  50297 
50379  50393  50406  50420  50433 

49900  49914  49927  49941   49955 
50037  50051   50065   50079  50092 
50174  50188  50202  50215   50229 
50311   50325   50338  50352   50365 
50447  50461   50474  50488  50501 

320 

321 
322 
323 

324 

50515  50529  50542  50556  50569 
50651   50664  50678  50691   50705 
50786  50799  50813   50826  50840 
50920  50934  50947  50  961   50974 
51055   51068  51081  51095   51108 

50583  505%  50610  50623   50637 
50718  50732  50745   50759  50772 
50853   50866  50880  50893   50907 
50987  51001   51014  51028  51041 
51  121   51  135   51  148  51  162  51  175 

325 
326 
327 
328 

329 

51188  51202  51215   51228  51242 
51322  51335  51348  51362  51375 
51455   51468  51481   51495  51508 
51587  51601   51614  51627  51640 
51  720  51  733  51  746  51  759  51  772 

51255  51268  51282  51295   51308 
51388  51402  51415   51428  51441 
51521   51534  51548  51561  51574 
51654  51667  51680  51693   51706 
51786  51799  51812  51825   51838 

33O 

331 
332 
333 
334 

51851  51865   51878  51891   51904 
51983  519%  52009  52022  52035 
52114  52127  52140  52153   52166 
52244  52257  52270  52284  52297 
52375   52388  52401   52414  52427 

51917  51930  51943  51957  51970 
52048  52061   52075   52088  52101 
52179  52192  52205   52218  52231 
52310  52323  52336  52349  52362 
52440  52453   52466  52479  52492 

335 
336 
337 
338 
339 

52504  52517  52530  52543   52556 
52634  52647  52660  52673   52686 
52763  52776  52789  52802  52815 
52892  52905   52917  52930  52943 
53020  53033  53046  53058  53071 

52569  52582  52595   52608  52621 
52699  52711   52724  52737  52750 
52827  52840  52853   52866  52879 
52956  52969  52982  52994  53007 
53084  53097  53110  53122  53135 

340 

341 

342 
343 
344 

53148  53161  53173  53186  53199 
53275   53288  53301   53314  53326 
53403  53415   53428  53441   53453 
53529  53542  53555   53567  53580 
53656  53668  53681   53694  53706 

53212  53224  53237  53250  53263 
53339  53352  53364  53377  53390 
53466  53479  53491   53504  53517 
53593  53605  53618  53631   53643 
53719  53732  53744  53757  53769 

345 
346 
347 
348 
349 

53782  53794  53807  53820  53832 
53908  53920  53933   53945   53958 
54033  54045   54058  54070  54083 
54158  54170  54183  54195   54208 
54283   54295  54307  54320  54332 

53845  53857  53870  53882  53895 
53970  53983  53995  54008  54020 
54095   54108  54120  54133   54145 
54220  54233  54245   54258  54270 
54345  54357  54370  54382  54394 

350 

54407   54419  54432   54444   54456 

54469  54481   54494  54506  54518 

N 

O           1           2           3           4 

56789 

300-350 


350-400 


3 


54407  54419  54432  54444  54456 

54531  54543  54555  54568  54580 

54654  54667  54679  54691  54704 

54777  54790  54802  54814  54827 

54900  54913  54925  54937  54949 

55023  55035  55047  55060  55072 

55145  55157  55169  55182  55194 

55267  55279  55291  55303  55315 

55388  55400  55413  55  425.  55437 

55509  55522  55534  55546  55558 

55630  55642  55654  55666  55678 

55751  55763  55775  55787  55799 

55871  55883  55  895  55907  55919 

55991  56003  56015.  56027  56038 

56110  56122  56134  56146  56158 

56229  56241  56253  56265  56277 

56348  56360  56372  56384  563% 

56467  56478  56490  56502  56514 

56585  56597  56608  56620  56632 

56703  56744  56726  56738  56750 


56820  56832  56844 

56937  56949  56961 

57054  57066  57078 

57171  57  183  57194 

57287  57299  57310 

57403  57415  57426 

57519  57530  57542 

57634  57646  57657 

57749  57761  57772 

57864  57875  57887 


56855  56867 

56972  56984 

57  089  57101 

57206  57217 

57322  57334 

57438  57449 

57553  57565 

57669  57680 

57784  57795 

57898  57910 


57978  57990  58001  58013  58024 

58092  58104  58115  58127  58138 

58206  58218  58229  58240  58252 

58320  58331  58343  58354  58365 

58433  58444  58456  58467  58478 

58546  58557  58569  58580  58591 
58659  58670  58681  58692  58704 
58771  58782  58794  58805  58816 
58883  58894  58906  58917  58928 
58995  59006  59017  59028  59040 

59106  59118  59129  59140  59151 

59218  59229  59240  59251  59262 

59329  59340  59351  59362  59373 

59439  59450  59461  59472  59483 

59550  59561  59572  59583  59594 

59660  59671  59  682  59693  59704 

59770  59780  59791  59802  59813 

59879  59890  59901  59912  59923 

59988  59999  60010  60021  60032 

60097  60108  60119  60130  60141 

60206  60217  60228  60239  60249 


r, 


54481  54494  54506  54518 

54605  54617  54630  54642 

54728  54741  54753  54765 

54851  54864  54876  54888 

54974  54986.54998  55011 

55  0%  55  108  55  121  55  133 
55218  55230  55242  55255 
55340  55352  55364  55376 
55461  55473  55485  55497 
55582  55594  55606  55618 


54469 
54593 
54716 
54839 
54962 

55084 
55206 
55328 
55449 
55570 

55691   55703  55715  55727  55739 

55811  55823  55835  55847  55859 

55931   55943  55955  55967  55979 

56050  56062  56074  56086  56098 

56170  56182  56194  56205  56217 

56289  56301   56312  56324  56336 

56407  56419  56431   56443  56455 

56526  56538  56549  56561  56573 

56644  56656  56667  56679  56691 

56761  56773  56785  56797  56808 

56879  56891  56902  56914  56926 
56996  57008  57019  57031  57043 
57113  57124  57136  57148  57159 
57229  57241  57252  57264  57276 
57345  57357  57368  57380  57392 

57461  57473  57484  574%  57507 

57576  57588  57600  57611  57623 

57692  57703  57715  57726  57738 

57807  57818  57830  57841  57852 

57921  57933  57944  57955   57  %7 

58035  58047  58058  58070  58081 

58149  58161  58172  58184  58195 

58263  58274  58286  58297  58309 

58377  58388  58399  58410  58422 

58490  58501   58512  58524  58535 

58602  58614  58625  58636  58647 

58715  58726  58737  58749  58760 

58827  58838  58850  58861  58872 

58939  58950  58961  58973  58984 

59051   59062  59073  59084  59095 

59162  59173  59184  59195  59207 

59273   59284  59295   59306  59  318 

59384  59395   59406  59417  59428 

59494  59506  59517  59528  59539 

59605  59616  59627  59638  59649 

59715  59726  59737  59748  59759 
59824  59835  59846  59857  59S6S 
59934  59945  59956  59966  59977 
60043  60054  60065  60076  600S6 
60152  60163  60173  60184  60195 

60260  60271   602S2   60293   60304 


350-400 


400-450 


H 

01234 

56789 

400 
401 
402 
403 
404 

60206  60217  60228  60239  60249 
60314  60325   60336  60347  60358 
60423  60433  60444  60455   60466 
60531  60541   60552  60563  60574 
60638  60649   60660  60670  60681 

60260  60271  60282  60293  60304 
60369  60379  60390  60401   60412 
60477  60487  60498  60509  60520 
60584  60595  60606  60617  60627 
60692  60703  60713  60724  60735 

405 
406 
407 
408 
409 

60746  60756  60767  60778  60788 
60853  60863   60874  60  885   60895 
60959  60970  60981   60991  61002 
61066  61077  61087  61098  61109 
61172  61183  61194  61204  61215 

60799  60810  60821  60831   60842 
60906  60917  60927  60938  60949 
61013  61023  61034  61045  61055 
61  119  61  130  61  140  61  151  61  162 
61225  61236  61247  61257  61268 

410 

411 
412 
413 
414 

61278  61289  61300  61310  61321 
61384  61395  61405   61416  61426 
61490  61500  61511   61521   61532 
61595  61606  61616  61627  61637 
61700  61711  61721  61731  61742 

61331  61342  61352  61363  61374 
61437  61448  61458  61469  61479 
61542  61553  61563  61574  61584 
61648  61658  61669  61679  61690 
61752  61763  61773  61784  61794 

415 
416 
417 
418 
419 

61805  61815  61826  61836  61847 
61909  61920  61930  61941  61951 
62014  62024  62034  62045   62055 
62118  62128  62138  62149  62159 
62221  62232  62242  62252  62263 

61857  61868  61878  61888  61899 
61962  61972  61982  61993   62003 
62066  62076  62086  62097  62107 
62170  62180  62190  62201   62211 
62273  62284  62294  62304  62315 

42O 

421 
422 
423 
424 

62325  62335  62346  62356  62366 
62428  62439  62449  62459  62469 
62531  62542  62552  62562  62572 
62634  62644  62655  62665   62675 
62737  62747  62757  62767  62778 

62377  62387  62397  62408  62418 
62480  62490  62500  62511   62521 
62583   62593   62603   62613   62624 
62685  626%  62706  62716  62726 
62788  62798  62808  62818  62829 

425 
426 
427 
428 
429 

62839  62849  62859  62870  62880 
62941   62951   62961   62972   62982 
63043  63053  63063  63073   63083 
63144  63155  63165  63175  63185 
63246  63256  63266  63276  63286 

62890  62900  62910  62921   62931 
62992  63002  63012  63022  63033 
63094  63104  63114  63124  63134 
63195   63205  63215  63225  63236 
63296  63306  63317  63327  63337 

43O 

431 
432 
433 

434 

63347  63357  63367  63377  63387 
63448  63458  63468  63478  63488 
63548  63558  63568  63579  63589 
63649  63659  63669  63679  63  689 
63  749  63  759  63  769  63  779  63  789 

63397  63407  63417  63428  63438 
63498  63508  63518  63528  63538 
63599  63609  63619  63629  63639 
63699  63709  63719  63729  63739 
63799  63809  63819  63829  63839 

435 
436 
437 
438 
439 

63849  63859  63869  63879  63  889 
63949  63959  63  969  63979  63988 
64048  64058  64068  64078  64088 
64147  64157  64167  64177  64187 
64246  64256  64266  64276  64286 

63899  63909  63919  63929  63939 
63998  64008  64018  64028  64038 
64098  64108  64118  64128  64137 
64197  64207  64217  64227  64237 
64296  64306  64316  64326  64335 

44O 

441 
442 
443 
444 

64345  64355  64365  64375   64385 
64444  64454  64464  64473   64483 
64542  64552  64562  64572  64582 
64640  64650  64660  64670  64  680 
64738  64748  64758  64768  64777 

64395   64404  64414  64424  64434 
64493   64503   64513   64523   64532 
64591   64601   64611   64621   64631 
64  689  64699  64709  64719  64729 
64787  64797  64807  64816  64826 

445 
446 
447 
448 
449 

64836  64846  64856  64865  64875 
64933   64943   64953   64963   64972 
65031  65040  65050  65060  65070 
65128  65137  65147  65157  65167 
65225  65234  65244  65254  65263 

64885   64895   64904  64914   64924 
64  982   64992  65002  65011   65021 
65079  65089  65099  65108  65118 
65  176  65  186  65  196  65  205  65  215 
65273  65283  65292  65302  65312 

45O 

65321   65331   65341   65350  65360 

65369  65379  65389  65398  65408 

N 

O           1           2           3           4 

56789 

400-450 


450-500 


O 


3 


o 


8 


497 
498 


65321  65331  65341  65350  65360 

65418  65427  65437  65447  65456 

65514  65523  65533  65543  65552 

65610  65619  65629  65639  65648 

65  706  65  715  65  725  65  734  65  744 

65801  65811  65820  65830  65839 
65896  65906  65916  65925  65935 
65992  66001  66011  66020  66030 
66087  66096  66106  66115  66124 
66181  66191  66200  66210  66219 

66276  66285  66295  66304  66314 

66370  66380  66  389  66398  66408 

66464  66474  664S3  66492  66502 

66558  66567  66577  66586  665% 

66652  66661  66671  66680  66689 

66745  66755  66764  66773  66783 

66839  66848  66857  66867  66876 

66932  66941  66950  66960  66969 

67025  67034  67043  67052  67062 

67117  67127  67136  67145  67154 

67210  67219  67228  67237  67247 

67302  67311  67321  67330  67339 

67394  67403  67413  67422  67431 

67486  67495  67504  67514  67523 

67578  67587  67596  67605  67614 

67669  67679  67688  67697  67706 
67761  67770  67779  67788  67797 
67852  67861  67870  67879  67888 
67943  67952  67  961  67970  67979 
68034  68043  68052  68061  68070 

68124  68133  68142  68151  68160 

68215  68224  68233  68242  68251 

68305  68314  68323  68332  68341 

68395  68404  68413  68422  68431 

68485  68494  68502  68511  68520 

68574  68583  68592  68601  68610 

68664  68673  68681  68690  68699 

68753  68762  68771  68780  68789 

68842  68851  68860  68869  68878 

68931  68940  68949  68958  68966 

69020  69028  69037  69046  69055 

69108  69117  69126  69135  69144 

69197  69205  69214  69223  69232 

69285  69294  69302  69311  69320 

69373  69381  69390  69399  69408 


69461  69469 
69548  69557 
69636  69644 
69723  69732 
69810  69819 

69897   69906 


69478  69487  694% 

69566  69574  69583 

69653  69662  69671 

69740  69749  69758 

69827  69836  69845 

69914  69923  69932 


65369  65379  65389  65398  65408 

65466  65475   65485  65495   65504 

65562  65571   65581  65591  65600 

65658  65667  65677  65686  656% 

65753  65763  65772  65782  65792 

65849  65858  65868  65877  65887 

65944  65954  65963  65973  65982 

66039  66049  66058  66068  66077 

66134  66143   66153  66162  66172 

66229  66238  66247  66257  66266 

66323  66332  66342  66351  66361 

66417   66427  66436  66445   66455 

66511  66521  66530  66539  66549 

66605   66614  66624  66633  66642- 

66699  66708  66717  66727  66736 

66792  66801  66811  66820  66829 
66885  66894  66904  66913  66922 
66978  66987  66997  67006  67015 
67071  67080  67089  67099  67108 
67164  67173  67182  67191  67201 

67256  67265  67274  67284  67293 
67348  67357  67367  67376  67385 
67440  67449  67459  67468  67477 
67532  67541  67550  67560  67569 
67624  67633  67642  67651  67660 

67715  67724  67733  67742  67752 
67806  67815  67825  67834  67843 
67897  67906  67916  67925  67934 
67988  67997  68006  680i5  68024 
68079  68088  68097  68106  68115 

68169  68178  68187  681%  68205 
68260  68269  68278  68287  68296 
68350  68359  68368  68377  68386 
68440  68449  68458  68467  68476 
68529  68538  68547  68556  68565 

68619  68628  68637  68646  68655 

68708  68717  68726  68735  68744 

68797  68806  68815  68824  68833 

68886  68895  68904  68913  68922 

68  975  68984  68993  69002  69011 

69064  69073  69082  69090  69099 
69152  69161  69170  69179  69188 
69241  69249  69258  69267  69276 
69329  69338  69346  69355  69364 
69417  69425  69434  69443  69452 

69504  69513  69522  69531  69539 
69592  69601  69609  69618  69627 
69679  69688  69697  69705  69714 
69767  69775  69784  69793  69801 
69854  69862  69871  69880  69888 
69940  69949  69958  69966  69975 


450  -  500 


10 


500-550 


N 

01234 

56789 

5OO 

501 
502 
503 
504 

69897  69906  69914  69923  69932 
69984  69992   70001   70010  70018 
70070  70079  70088  70096  70105 
70157  70165   70174  70183   70191 
70243   70252  70260  70269  70278 

69940  69949  69958  69966  69975 
70027  70036  70044  70053    70062 
70114   70122   70131    70140   70148 
70200  70209  70217   70226   70234 
70286  70295   70303   70312   70321 

505 
506 
507 
508 
509 

70329  70338  70346  70355    70364 
70415   70424  70432   70441   70449 
70501   70509  70518  70526  70535 
70586  70595   70603   70612   70621 
70672  70680  70689  70697  70706 

70372  70381   70389   70398  70406 
70458  70467   70475    70484  70492 
70544  70552  70561    70569  70578 
70629  70638   70646   70655   70663 
70714  70723   70731    70740  70749 

510 

511 
512 
513 
514 

70757  70766  70774  70783   70791 
70842  70851   70859  70868   70876 
70927  70935   70944   70952  70961 
71012   71020  71029   71037    71046 
71096  71105   71113   71122   71130 

70800  70808   70817  70825   70834 
70885   70893   70902    70910  70919 
70969  70978    70986  70995   71003 
71054   71063   71071    71079  71088 
71  139  71  147   71  155   71  164  71  172 

515 
516 
517 
518 
519 

71181   71189  71198   71206  71214 
71265   71273   71282  71290  71299 
71349  71357  71366  71374  71383 
71433   71441   71450  71458  71466 
71517  71525   71533   71542    71550 

71223   71231   71240   71248  71257 
71307   71315    71324  71332  71341 
71391   71399  71408   71416  71425 
71475   71483   71492    71500  71508 
71559  71567    71575    71584  71592 

520 

521 
522 
523 
624 

71600  71609  71617  71625   71634 
71684  71692   71700  71709  71717 
71767  71775   71784  71792  71800 
71850  71858  71867  71875    71883 
71933   71941   71950  71958  71966 

71642  71650  71659  71667   71675 
71  725   71  734  71  742   71  750  71  759 
71809  71817    71825   71834   71842 
71892  71900  71908   71917  71925 
71975   71983   71991   71999  72008 

525 
526 
527 
528 
529 

72016  72024   72032   72041   72049 
72099  72107  72115   72123   72132 
72181   72189  72198   72206  72214 
72263   72272  72280   72288   72296 
72346  72354   72362   72370   72378 

72057  72066  72074    72082  72090 
72140  72148  72156  72165   72173 
72222   72230    72239  72247   72255 
72304  72313    72321   72329  72337 
72387   72395   72403    72411    72419 

530 

531 

532 

533 
534 

72428  72436  72444   72452  72460 
72509  72518  72526  72534  72542 
72591   72599  72607  72616  7.2624 
72673   72681   72689  72697   72705 
72754  72762  72770  72779  72787 

72469  72477   72485    72493   72501 
72550  72558   72567  72575   72583 
72632    72640  72648  72656  72665 
72713   72722    72730  72738  72746 
72795   72803    72811   72819  72827 

535 
536 
537 
538 
539 

72835   72843   72852   72860  72  868 
72916  72925   72933   72941    72949 
72997  73006  73014  73022   73030 
73078  73086  73094   73102   73111 
73159  73167  73175  73183   73191 

72876  72884    72892  72900  72908 
72957   72  965    72973   72981   72989 
73038   73046  73054  73062   73070 
73119  73127   73135  73143   73151 
73199  73207    73215   73223   73231 

540 

541 
542 
543 
544 

73239  73247  73255   73263    73272 
73320  73328  73336  73344  73352 
73400  73408  73416  73424    73432 
73480  73488  734%  73504  73512 
73560  73568  73576  73584   73592 

73280   73288  73296  73304   73312 
73360  73368  73376  73384  73392 
73440  73448  73456  73464  73472 
73520  73528   73536  73544   73552 
73600  73608   73616  73624  73632 

545 
546 
547 
548 
549 

73640  73648  73656  73664  73672 
73719  73727  73735   73743   73751 
73799  73807  73815   73823   73830 
73878  73886  73894   73902   73910 
73957  73965   73973   73981   73989 

73679  73687   73695   73703    73711 
73759  73767    73775   73783   73791 
73838  73846  73854  73862   73870 
73918  73926  73933   73941   73949 
73997  74005   74013   74020   74028 

550 

74036  74044  74052  74060  74068 

74076  74084   74092   74099   74107 

N 

01234 

56789 

500-550 


550-600 


n 


0 


74036  74044  74052   74060  74068 

74115  74123  74131   74139  74147 

74194  74202  74210  74218  74225 

74273  74280  74  288  74296  74304 

74351  74359  74367  74374  74382 

74429  74437  74445  74453   74461 

74507  74515   74523  74531   74539 

74586  74593   74601  74609  74617 

74663  74671   74679  74687  74695 

74741  74749  74757  74764  74772 

74819  74827  74834  74842  7485.0 
74896  74904  74912  74920  74927 
74974  74981  74  989  74997  75005 
75051  75059  75066  75074  75082 
75 128  75136  75143  75151  75159 

75205  75213  75220  75228  75236 

75282   75289  75297  75305   75312 

75358  75366  75374  75381   75389 

75435   75442  75450  75458  75465 

75511   75519  75526  75534   75542 

75587  75595   75603   75610  75618 

75664  75671   75679  75686  75694 

75740  75747  75755  75762  75770 

75815  75823   75831   75838  75846 

75891  75899  75906  75914  75921 

75  967  75974  75982  75989  75997 

76042  76050  76057   76065   76072 

76118  76125  76133   76140  76148 

76193  76200  76208  76215   76223 

76268  76275  76283   76290  76298 

76343  76350  76358  76365   76373 

76418  76425  76433   76440  76448 

76492  76500  76507   76515   76522 

76567  76574  76582  76589  76597 

76641  76649  76656  76664  76671 

76716  76723  76730  76738  76745 

76790  76797  76805  76812  76819 

76864  76871  76879  76886  76893 

76938  76945  76953   76960  76967 

77012  77019  77026  77034   77041 

77085  77093  77100  77107  77115 

77159  77166  77173  77181  77188 

77232   77240  77247  77254  77262 

77305   77313  77320  77327  77335 

77379  77386  77393  77401  77408 


77452  77459 

77525  77532 

77597  77605 

77670  77677 

77743  77750 

77815  77822 


77466  77474  77481 

77539  77546  77554 

77612   77619  77627 

77685   77692  77699 

77757  77764  77772 

77830  77837  77844 


74076  74084  74092  74099  74107 

74155   74162  74170  74178  74186 

74233   74241   74249  74257  74265 

74312   74320  74327  74335  74343 

74390  74398  74406  74414  74421 

74468  74476  74484  74492  74500 

74547   74554  74562  74570  74578 

74624  74632   74640  74648  74656 

74702   74710  74718  74726  74733 

74780  74788  747%  74803  74811 

74858  74865   74873  74881  74889 

74935   74943   74950  74958  74966 

75012  75020  75028  75035  75043 

75089  75097  75105  75113  75120 

75166  75174  75182  75189  75197 

75243  75251   75259  75266  75274 

75320  75328  75335  75343  75351 

75397  75404   75412  75420  75427 

75473   75481   75488  75496  75504 

75549  75557  75565  75572  75580 

75626  75633  75641  75648  75656 

75702   75709  75717  75724  75732 

75778  75785   75793  75800  75808 

75853  75861   75868  75876  75884 

75929  75937  75944  75952  75959 

76005  76012   76020  76027  76035 

76080  76087  76095  76103  76110 

76155   76163   76170  76178  76185 

76230  76238  76245  76253  76260 

76305   76313   76320  76328  76335 

76380  76388  76395  76403  76410 

76455   76462   76470  76477  76485 

76530  76537  76545  76552  76559 

76604  76612   76619  76626  76634 

76678  76686  76693  76701  76708 

76753   76760  76768  76775  76782 

76827  76834  76842  76849  76856 

76901   76908  76916  76923  76930 

76975   76982   76989  76997  77004 

77048  77056  77063  77070  77078 

77122  77129  77137  77144  77151 

77195   77203   77210  77217  77225 

77269  77276  77283  77291  77298 

77342  77349  77357  77364  77371 

77415   77422  77430  77437  77444 


77488 
77561 
77634 
77706 

77779 


77495 
77568 
77641 
77714 
77786 


77503  77510  77517 

77576  77583  77590 

77648  77656  77663 

77721  77728  77735 

77793  77801  77808 


77851  77859  77866  77873  77880 


550-600 


12 


600-650 


N 

01234 

56789 

600 
6C1 
602 
603 
604 

77815   77822  77830  77837  77844 
77887   77895   77902   77909   77916 
77960  77967  77974  77981  77988 
78032   78039   78046  78053   78061 
78104  78111    78118  78125   78132 

77851   77859  77866  77873    77880 
77924  77931   77938   77945    77952 
779%  78003   78010    78017   78025 
78068  78075   78082    7SOS9  78097 
78140  78147   78154   78161   78168 

605 
606 
607 
608 
609 

78176  78183   78190  78197  78204 
78247   78254  78262    78269  78276 
78319  78326  78333   78340  78347 
78390  78398  78405    78412   78419 
78462  78469  78476   78483   78490 

78211   782J9   78226  78233   78240 
78283   78290  78297   78305   78312 
78355   78362   78369  78376  78  383 
78426  78433   78440  78447  78455 
78497   78504  78512  78519  78526 

610 

611 
612 

613 
614 

78533   78540  78547   78554  78561 
78604  78611   78618   78625   78633 
78675   78682   78689   786%  78704 
78746  78753   78760   78767  78774 
78817  78824   78831   78838  78845 

78569  78576  78583   78590  78597 
78640  78647    78654  78661   78  668 
78711   78718  78725   78732   78739 
78  781   78789   78  796   78  803   78810 
78S52  78859  78866  78  873   78880 

615 
616 
617 
618 
619 

78  888  78895   78902   78909   78916 
78958   78  965   78972   78  979  78986 
79029  79036  79043   79050   79057 
79099  79106  79113   79120  79127 
79169  79176  79183  79190  79197 

78923   78930  78937  78944  78951 
78993   79000  79007  79014   79021 
79064  79071   7907S  79085   79092 
79134   79141   79148  79155   79162 
79204  79211   79218  79225   79232 

62O 

621 
622 
623 
624 

79239  79246  79253   79260  79267 
79309  79316  79323   79330  79337 
79379  79386  79393   79400  79407 
79449   79456  79463   79470  79477 
79518  79525   79532  79539  79546 

79274  79281    79288  79295   79302 
79344   79351   79358   79365    79372 
79414   79421   7942S  79435   79442 
79  484   79491   79498  79505   79511 
79553   79560   79567   79574   79581 

625 
626 
627 
628 
629 

79588  79595   79602   79609  79616 
79657  79664   79671   79678  79685 
79727  79734  79741   79748  79754 
797%  79803   79S10  79817  79824 
79865   79872   79879  79886  79893 

79623   79630  79637  79644   79650 
79692   79699  79706  79713   79720 
79761   79768  79775   79  782  79  789 
79831    79837  79844  79851   79  858 
79900  79906  79913   79920  79927 

630 

631 
632 
633 
634 

79934  79941   79948   79955   79  %2 
80003   80010  80017  80024  80030 
80072  80079  80  085   SO  092  80099 
80140  80147  80154  80161   80168 
80209  80216  80223  80229  80236 

79969  79975   79982   79  989  79996 
80037  80044  80051   80058  SO  065 
80106  80113  SO  120  SO  127  80134 
80175   S01S2  S01SS  SO  195   80202 
80243   80250  80257  80264  SO  271; 

635 
636 
637 
638 
639 

80277  80284  80291   80298  80305 
80346  80353  SO  359  80366  SO  373 
SO  414  80421   80428  80434  80441 
SO  482  804S9  80496  80502  SO  509 
SO  550  80557  80564  SO  570  SO  577 

SO  312  SO  318  80325   80332  SO  339 
S03SO  SO  387  80393   80400  SO  407 
S044S  80455   80462  80468  804751 
80516  SO  523   SO  530  80536  80  543  ; 
SO  584  80591   80598  80604  8061H 

640 

641 
642 
643 
644 

80618  SO  625   80632  8063S  SO  645 
S06S6  SO  693   SO  699  80706  SO  713 
80754  80760  SO  767  80774  SO  781 
80  821   80828  80835   80  841   80S4S 
S08S9  SO  895   80902  80909  80916 

80652  80659  80665   80672  8067» 
80720  80726  80733  80740  SO  747] 
80787  80794  SO  801   SO  SOS  SO  814 
80855   SOS62  SOS6S  SO  875   SOSS2 
SO  922  80929  SO  936  80943   SO  949  j 

645 

646 
647 
648 
649 

80956  SO  963   80969  SO  976  80983 
81023   81030  81037  81043   81050 
81090  SI  097  81104  81111   81117 
81158  81164  81  171   81  17S  81184 
81224  81231   81238  81245   81251 

80990  80996  SI  003   81010  SI  017 
81057  81064  81070  SI  077  S10S4; 
81124  81131   81137  81144  81  ISM 
81191   81  198  81204  SI  211   SI  218 
SI  258  81265   81271   81278  81285. 

630 

81291   S129S   SI  305   81  311   SI  318 

SI  32£   SI  331   S133S  81345   SI  351 

M 

01234 

o           G           7           8           9 

600-650 


650-700 


13 


3 


81291  81298  81305  81311  81318 

81358  81365  81371  81378  81385 

81425  81431  81438  81445  81451 

81491  81498  81505  81511  81518 

81558  81564  81571  81578  81584 

81624  81631  81637  81644  81651 

81690  81697  81704  81710  81717 

81757  81763  81770  81776  81783 

81823  81829  81836  81842  81849 

81889  81895  81902  81908  81915 

81954  81 961  81968  81974  81981 

82020  82027  82033  82040  82046 

82086  82092  82099  82105  82112 

82151  82158  82164  82171  82178 

82217  82223  82230  82236  82243 

82282  82289  82295  82302  82308 

82347  82354  82  360  82367  82373 

82413  82419  82426  82432  82439 

82478  82484  82491  82497  82504 

82543  82549  82556  82562  82569 

82607  82614  82620  82627  82633 

82672  82679  82685  82692  82698 

82737  82743  82750  82756  82763 

82802  82808  82814  82821  82827 

82866  82872  82879  82885  82892 

82930  82937  82943  82950  82956 

82995  83001  83008  83014  83020 

83059  83065  83072  83078  83085 

83123  83129  83136  83142  83149 

83187  83193  83200  83206  83213 

83251  83257  83264  83270  83276 

83315  83321  83327  83334  83340 

83378  83385  83391  83398  83404 

83442  83448  83455  83461  83467 

83506  83512  83518  83525  83531 

83569  83575  83582  83588  83594 

83632  83639  83645  83651  83658 

83696  83702  83708  83715  83721 

83759  83765  83771  83778  83784 

83822  83828  83835  83841  83847 

83885  83891  83897  83904  83910 

83948  83954  83960  83967  83973 

84011  84017  84023  84029  84036 

84073  84080  84086  84092  84098 

84136  84142  84148  84155  84161 


84198  84205  84211 

84261  84267  84273 

84323  84330  84336 

84386  84392  84398 

84448  84454  84460 


84217 
84280 
84342 
84404 
84466 


84223 
84286 
84348 
84410 
84473 


81325  81331  81338  81345  81351 

81391  81398  81405  81411  81418 

81458  81465  81471  81478  81485 

81525  81531  81538  81544  81551 

81591  81598  81604  81611  81617 


81657 
81723 
81790 
81856 
81921 

81987 
82053 
82119 
82184 
82249 


81664 
81730 
817% 
81862 
81928 

81994 
82060 
82125 
82191 
82256 


81671 
81737 
81803 
81869 
81935 

82000 
82066 
82132 
82197 
82263 


81677  81684 
81743  81750 
81809  81816 
81875  81882 
81941  81948 

82007  82014 
82073  82079 
82138  82145 
82204  82210 
82269  82276 


84510  84516  84522  84528  84535 


82315  82321  82328  82334  82341 

82380  82387  82393  82400  82406 

82445  82452  82458  82465  82471 

82510  82517  82523  82530  82536 

82575  82582  82588  82595  82601 

82640  82646  82653  82659  82666 

82705  82711  82718  82724  82730 

82769  82776  82782  82789  82795 

82834  82840  82847  82853  82860 

82898  82905  82911  82918  82924 

82963  82  969  82975  82982  82988 

83027  83033  83040  83046  83052 

83091  83097  83104  83110  83117 

83155  83161  83168  83174  83181 

83219  83225  83232  83238  83245 

83283  83289  83296  83302  83308 

83347  83353  83359  83366  83372 

83410  83417  83423  83429  83436 

83474  83480  83487  83493  83499 

83537  83544  83550  83556  83563 

83601  83607  83613  83620  83626 

83664  83670  83677  83683  83689 

83727  83734  83740  83746  83753 

83790  83797  83803  83809  83816 

83853  83860  83866  83872  83879 

83916  83923  83929  83935  83942 

83979  83985  83992  83998  84004 

84042  84048  84055  84061  84067 

84105  84111  84117  84123  84130, 

84167  84173  84180  84166  84192 

84230  84236  84242  84248  84255 

84292  84298  84305  84311  84317 

84354  84361  84367  84373  84379 

84417  84423  84429  84435  84442 

84479  84485  84491  84497  84504 

84541  84547  84553  84559  84566 


650-700 


14 


700-750 


N 

01234 

56789 

TOO 
701 
702 
703 
704 

84510  84516  84522   84528   84  535. 
84572  84578  84584  84590  84597 
84634  84640  84646  84652  84  658 
846%  84702  84708  84714  84720 
84757  84763  84770  84776  84782 

84541   84547  84  553   84559  84566 
84603   84609   84  615   84  621   84628 
84665  84671  84677  84683  84689 
84726  84733   84739  84745   84751 
84788  84794  84800  84807  84813 

705 
706 
707 
708 
709 

84819  84825  84S31   84837  84844 
84880  84887  84893   84899  84905 
84942  84948  84954  84960  84  967 
85003   85009  85016  85022  85  028 
85065  85071   85077  85083  85089 

84850  84856  84862  84868  84874 
84911  84917  84924  84930  84936 
84973   84979  84985   84991   84997 
85  034  85  040  85  046  85052  85  058 
85095   85101   85107  85114  85120 

71O 

711 
712 
713 
714 

85126  85132  85  138  85  144  85150 
85187  85193   85199  85205   85211 
85248  85254  85260  85266  85272 
85309  85315   85321   85327  85333 
85370  85376  85382  85388  85394 

85156  85163   85169  85175  85181 
85217  85224  85230  85236  85242 
85278  85285   85291   85297  85303 
85339  85345   85352  85358  85364 
85400  85406  85412  85418   85425 

715 
716 
717 
718 
719 

85431  85437  85443  85449  85455 
85491   85497  85503   85509  85516 
85552  85558  85564  85570  85576 
85612  85618  85625  85631   85637 
85673  85679  85685  85691   85697 

85461   85467  85473   85479  85485 
85522  85528  85534  85540  85546 
85582  85588  85594  85600  85606 
85643  85649  85655   85661   85667 
85703   85709  85715  85721   85727 

720 

721 

722 
723 
724 

85733  85739  85745  85751  85757 
85794  85800  85806  85  812  85818 
85854  85860  85866  85872  85878 
85914  85920  85926  85932  85938 
85974  85980  85986  85992  85998 

85763  85769  85775  85781  85788 
85824  85830  85836  85842  85848 
85884  85890  858%  85902  85908 
85944  85950  85956  85962  85968 
86004  86010  86016  86022  86028 

725 
726 
727 
728 

729 

86034  86040  86046  86052  86058 
86094  86100  86106  86112  86118 
86153  86159  86165   86171   86177 
86213  86219  86225   86231  86237 
86273  86279  86285  86291   86297 

86064  86070  86076  86082  86088 
86124  86130  86  136  86141   86147  ! 
86183   86189  86  195   86201   86207 
86243   86249  86255   86261   86267 
86303   86308  86314  86320  86326 

73O 

731 
732 
733 

734 

86332  86338  86344  86350  86356 
86392  86398  86404  86410  86415 
86451  86457  86463  86469  86475 
86510  86516  86522  86528  86  534 
86570  86576  86581   86587  86593 

86362  86368  86374  86380  86386 
86421   86  427  86433   86439  86  445 
86481   86487  86493   86499  86504 
86540  86546  86  552  86  558  86564 
86599  86605   86611   86617  86623 

735 
736 
737 
738 

739 

86629  86635  86641  86646  86652 
86688  86694  86700  86705   86711 
86747  86753  86759  86764  86770 
86806  86812  86817  86823   86829 
86864  86870  86876  86882  86888 

86658  86664  86670  86676  86682 
86  717  86  723   86  729  86  735   86  741 
86776  86  782  86  7SS  86794  86SOO 
86835   86841   86  847  86853   86859 
86894  86900  86906  86911  869171 

740 

741 
742 
743 
744 

86923  86929  86935  86941  86947 
86982  86988  86994  86999  87005 
87040  87046  87052  87058  87064 
87099  87105  87111  87116  87122 
87157  87163  87169  87175   87181 

86953  86958  86964  86970  86976 
87011  87017  87023   87029  87035 
87070  87075   87081   87087  87093 
87128  87134  87140  87146  871511 
87186  87192  87198  87204  87210 

745 
746 
747 
748 
749 

87216  87221   87227  87233  87239 
87274  87280  87286  87291   87297 
87332  87338  87344  87349  87355 
87390  873%  87402  87408  87413 
87448  87454  87460  87466  87471 

87245  87251  87256  87262  87268 
87303  87309  87315  87320  87326 
87361  87367  87373  87379  87  384 
87419  87425  87431   87437  87  442 
87477  87483  87489  87495  87500  : 

750 

87506  87512  87518  87523  87529 

87535   87541   87547  87552  87558 

N 

O           1           2           3           4 

56789 

700-750 


750-800 


15 


8    9 


775 
770 
777 
770 
779 

780 
781 
782 
783 

784 

785 
786 
787 
788 
789 

79O 

791 
792 

793 
794 
795 
796 
797 
798 
799 


87506  87512  87518  87523  87529 

87564  87570  87576  87581  87587 

87622  87628  87633  87639  87645 

87679  87685  87691  87697  87703 

87737  87743  87749  87754  87760 

87795  87800  87806  87812  87818 

S7S52  87858  87864  87869  87875 

87910  87915  87921  87927  87933 

87  967  87973  87978  87984  87990 
88024  88030  88036  88041  88047 

880S1  88  087  88093  88098  88104 

88138  88144  88150  88156  88161 

88195  88  201  88207  88213  88218 

88252  88  258  88264  88270  88275 

88309  88315  88321  88  326  88332 

88366  88372  88377  88383  88389 

88423   88429  88434  88440  88446 

88  480  88485  88491  88497  88502 
88536  88542  88547  88553  88559 
88593  88598  88604  88610  88615 

88649  88655  88660  88666  88672 

88705   88711  88717  88722  88728 

88762  88767  88773  88779  88784 

88818  88824  88829  88835  88840 

88874  88880  8SSSS  88891  SS897 


88930 
88986 
89  042 
89  098 
89154 

89209 
89265 
89321 
89  3  76 
89432 


88936  88941 
88992  88997 
89048  89053 
89104  89109 
89159  89165 

89215  89221 
89271  89276 
89326  89332 
89382  89387 
89437  89443 


88947  88953 
89003  89009 
89059  89064 
89115  89120 
89170  89176 

89226  89232 
89282  89287 
89337  89343 
89393  89398 
89448  89454 


89487  89492  89498  89504  89509 

89542  89548  89553   89559  89564 

89597  89603   89609  89614  89620 

89653   89658  89664  89669  896/5 

89708  89713  89719  89724  89730 

89763  89768  89774  89779  89785 

89818  89823   89829  89834  89840 

89873   89878  89883   89889  89894 

89927  89933   89938  89944  89949 

89982  89988  89993   89998  90004 

90037  90042  90048  90053  90059 

90091  90097  90102  90108  90113 

90146  90151   90157  90162  90168 

90200  90206  90211  90217  90222 

90255  90260  90266  90271  90276 

90309  90314  90320  90325  90331 


87535  87541  87547  87552  87558 

87593  87599  87604  87610  87616 

87651  87656  87662  87668  87674 

87708  87714  87720  87726  87731 

87766  87772  87777  87783  87789 

87823  87829  87835  87841  87846 

87881  87887  87892  87898  87904 

87938  87944  87950  87955   87  961 

879%  88001  88007  88013  88018 

88053  88058  88064  88070  88076 

88110  88116  88121  88127  88133 
88167  88173  88178  88184  88190 
88224  88230  88235  88241  88247 
88281  88287  88292  88298  88304 
88338  88343  88349  88355  88360 

88395  88400  88406  88412  88417 
88451  88457  88463  88468  88474 
88508  88513  88519  88525  88530 
88564  88570  88576  88581  88587 
88621  88627  88632  88638  88643 

88677  88683  88689  88694  88700 

88734  88739  88745  88750  88756 

88790  88795  88801  88807  88812 

88846  88852  88857  88863  88868 

88902  88908  88913  88919  88925 

88958  88964  88  969  88975  88981 

89014   89020  89025  89031  89037 

89070  89076  89081  89087  89092 

89126  89131   89137  89143  89148 

89182  89187  89193  89198  89204 

89237  89243  89248  89254  89260 
89293  89298  89304  89310  89315 
89348  89354  89360  89365  89371 
89404  89409  89415  89421  89426 
89459  89465  89470  89476  89481 

89515  89520  89526  89531  89537 
89570  89575  89581  89586  89592 
89625  89631  89636  89642  89647 
89680  89686  89691  89697  89702 
89735  89741  89746  89752  89757 

89790  897%  89801  89807  89812 

89845   89851  89856  89862  89867 

89900  89905  89911  89916  89922 

89955   89960  89966  89971   89977 

90009  90015  90020  90026  90031 

90064  90069  90075  90080  90086 
90119  90124  90129  90135  90140 
90173  90179  90184  90189  90195 
90227  90233  90238  90244  90249 
90282  90287  90293  90298  90304 
90336  90342  90347  90352  90358 


8 


750-800 


16 


800-850 


N 

01234 

56789 

8OO 

801 
802 
803 
804 

90309  90314  90320  90325   90331 
90363  90369  90374  90380  90385 
90417  90423  90428  90434  90439 
90472  90477  90482  90488  90493 
90526  90531  90536  80542  90547 

90336  90342  90347  90352  90358 
90390  90396  90401  90407  90412 
90445  90450  90455   90461  90466 
90499  90504  90509  90515  90520 
90553  90558  90563   90569  90574 

805 
806 
807 
808 
809 

90580  90585  90590  90596  90601 
90634  90639  90644  90650  90655 
90687  90693  90698  90703  90709 
90741  90747  90752  90757  90763 
90795  90800  90806  90811  90816 

90607  90612  90617  90623  90628 
90660  90666  90671   90677  90682 
90714  90720  90725  90730  90736 
90768  90773  90779  90784  90789 
90822  90827  90832  90838  90843 

81O 

811 
812 
813 
814 

90849  90854  90859  90865  90870 
90902  90907  90913  90918  90924 
90956  90961  90966  90972  90977 
91009  91014  91020  91025  91030 
91062  91068  91073  91078  91084 

90875  90881  90886  90891  90897 
90929  90934  90940  90945  90950 
90982  90988  90993  90998  91004 
91036  91041  91046  91052  91057 
91089  91094  91100  91105  91110 

815 
816 
817 
818 
819 

91116  91121  91126  91132  91137 
91169  91174  91180  91185  91190 
91222  91228  91233  91238  91243 
91275  91281  91286  91291  91297 
91328  91334  91339  91344  91350 

91142  91148  91153  91158  91164 
91196  91201  91206  91212  91217 
91249  91254  91259  91265  91270 
91302  91307  91312  91318  91323 
91355  91360  91365  91371  91376 

820 

821 
822 
823 

824 

91381  91387  91392  91397  91403 
91434  91440  91445  91450  91455 
91487  91492  91498  91503  91508 
91540  91545  91551  91556  91561 
91593  91598  91603  91609  91614 

91408  91413  91418  91424  91429 
91461  91466  91471   91477  91482 
91514  91519  91524  91529  91535 
91566  91572  91577  91582  91587 
91619  91624  91630  91635  91640 

825 
826 
827 
828 
829 

91645  91651  91656  91661  91666 
91698  91703  91709  91714  91719 
91751  91756  91761  91766  91772 
91803  91808  91814  91819  91824 
91855  91861  91866  91871  91876 

91672  91677  91682  91687  91693 
91724  91730  91735  91740  91745 
91777  91782  91787  91793  91798 
91829  91834  91840  91845   91850 
91882  91887  91892  91897  91903 

830 

831 
832 
833 
834 

91908  91913  91918  91924  91929 
91960  91965  91971  91976  91981 
92012  92018  92023  92028  92033 
92065  92070  92075  92080  92085 
92117  92122  92127  92132  92137 

91934  91939  91944  91950  91955 
91986  91991   91997  92002  92007 
92038  92044  92049  92054  92059 
92091   92096  92101  92106  92111 
92143  92148  92153  92158  92163 

835 
836 
837 
838 
839 

92169  92174  92179  92184  92189 
92221  92226  92231  92236  92241 
92273  92278  92283  92288  92293 
92324  92330  92335  92340  92345 
92376  92381  92387  92392  92397 

92195  92200  92205   92210  92215 
92247  92252  92257  92262  92267 
92298  92304  92309  92314  92319 
92350  92355  92361  92366  92371 
92402  92407  92412  92418  92423 

840 

841 
842 
843 
844 

92428  92433  92438  92443  92449 
92480  92485  92490  92495  92500 
92531  92536  92542  92547  92552 
92583  92588  92593  92598  92603 
92634  92639  92645  92650  92655 

92454  92459  92464  92469  92474 
92505  92511  92516  92521  92526 
92557  92562  92567  92572  92578 
92609  92614  92619  92624  92629 
92660  92665   92670  92675  92681 

845 
846 
847 
848 
849 

92686  92691  92696  92701  92706 
92737  92742  92747  92752  92758 
92788  92793  92799  92804  92809 
92840  92845  92850  92855   92860 
92891  92896  92901  92906  92911 

92711   92716  92722  92727  92732. 
92763  92768  92773  92778  92  783 
92814  92819  92824  92829  92834 
92865  92870  92875   92881   92886 
92916  92921  92927  92932  92937 

850 

92942  92947  92952  92957  92  962 

92  967  92973  92978  92983  92  988 

N 

01234 

56789 

800-850 


850-900 


17 


3 


92942  92947  92952  92957  92962 

92993  92998  93003  93  008  93013 

93044  93049  93054  93059  93064 

93095  93100  93105  93110  93115 

93  146  93  151  93  156  93  16!  03  166 

93197  93202  93207  93212  93217 

93247  93252  93258  93263  93268 

93298  93303  93308  93313  93318 

93349  93354  93359  93364  93369 

93399  93404  93409  93414  93420 

934^0  93455  93460  93465  93470 

93500  93505  93510  93515  93520 

93551  93556  93561  93566  93571 

93601  93606  93611  93616  93621 

93651  93656  93661  93666  93671 

93702  93707  93712  93717  93722 

93752  93757  93762  93767  93772 

93802  93807  93  812  93  81 7  93822 

93852  93857  93862  93867  93  872 

93902  93907  93912  93917  93922 

93952  93957  93962  93967  93972 

94002  94007  94012  94017  94022 

94052  94057  94062  94067  94072 

94101  94106  94111  94116  94121 

94151  94156  94161  94166  94171 

94201  94206  94211  94216  94221 

94250  94255  94260  94265  94270 

94300  94305  94310  94315  94320 

94349  94354  94359  94364  94369 

94399  94404  94409  94414  94419 

94448  94453  94458  94463  94468 

94498  94503  94507  94512  94517 

94547  94552  94557  94562  94567 

945%  94601  94606  94611  94616 

94645  94650  94655  94660  94665 

94694  94699  94704  94709  94714 

94743  94748  94753  94758  94763 

94792  94797  94802  94807  94812 

94841  94846  94  851  94856  94861 

94890  94895  94900  94905  94910 

94939  94944  94949  94954  94959 

94  988  949V3  94998  95002  95007 
95036  95041  v5  046  95051  95056 
95085  95090  95  005  95100  95105 
95134  95139  95  1*3  95148  95153 


95182 
95231 
95279 
95328 
95376 


95187 
95236 
95  284 
95332 
95381 


95192  95197  95202 
95240  95245  95250 
95289  95294  95299 
95337  95342  95347 
95386  95390  95395 


92967  92973  92978  929S3  92988 

93018  93024  93029  93034  93039 

93069  93075  93080  93085  93090 

93120  93125  93131  93136  93141 

93171  93176  93181  93186  93192 


93222 
93273 
93323 
93374 
93425 

93475 
93526 
93576 
93626 
93676 


93227 
93278 
93328 
93379 
93430 

93480 
93531 
93581 
93631 
93682 


93232  93237  93242 
93283  93288  93293 
93334  93339  93344 
93384  93389  93394 
93435  93440  93445 

93485  93490  93495 

93536  93541  93546 

93586  93591  93596 

93636  93641  93646 

93687  93692  93697 


95424  95429  95434  95439  95444 


93727  93732  93737  93742  93747 

93777  93782  93787  93792  93797 

93827  93832  93837  93842  93847 

93877  93882  93887  93892  93897 

93927  93932  93937  93942  93947 

93977  93982  93987  93992  93997 
94027  94032  94037  94042  94047 
94077  94082  94086  94091  94096 
94126  94131  94136  94141  94146 
94176  94181  94186  94191  941% 

94226  94231  94236  94240  94245 
94275  94280  94285  94290  94295 
94325  94330  94335  94340  94345 
94374  94379  94384  94389  94394 
94424  94429  94433  94438  94443 

94473  94478  94483  94488  94493 
94522  94527  94532  94537  94542 
94571  94576  94581  94586  94591 
94621  94626  94630  94635  94640 
94670  94675  94680  94685  94689 

94719  94724  94729  94734  94738 
94768  94773  94778  94783  94787 
94817  94822  94827  94832  94836 

94  866  94  871-  94  876  94  880  94  885 
94915  94919  94924  94929  94934 

94963  94968  94973  94978  94983 
95012  95017  95022  95027  95032 
95061  95066  95071  95075  95  080 
95109  95114  95119  95124  95129 

95  158  95  163  95  168.  95  173  95  177 

95207  95211  95216  95221  95226 
95255  95260  95265  95270  95274 
95303  95308  95313  95318  95323 
95352  95357  95361  95366  95371 
95400  95405  95410  95415  95419 

95448  95453  95458  95463  95  468 


8 


859-900 


18 


900-950 


N 

O           1           2           3           4 

56789 

900 

901 
902 
903 
904 

95424  95429  95434  95439  95444 
95472  95477  95482  95487  95492 
95521  95525  95530  95535  95540 
95569  95574  95578  95583  95588 
95617  95622  95626  95631  95636 

95448  95453  95458  95463  95468 
95497  95501  95506  95511  95516 
95545  95550  95554  95559  95564 
95593  95598  95602  95607  95612 
95641  95646  95650  95655  95660 

905 
906 
907 
908 
909 

95665  95670  95674  95679  95684 
95713  95718  95722  95727  95732 
95761  95766  95770  95775  95780 
95809  95813  95818  95823  95828 
95856  95861  95866  95871  95875 

95689  95694  95698  95703  95708 
95  737  95  742  95  746  95  751  95  756 
95  785  95  789  95  794  95  799  95  804 
95832  95837  95842  95847  95852 
95880  95885   95890  95895  95899 

910 

911 
912 
913 

914 

95904  95909  95914  95918  95923 
95952  95957  95961   95966  95971 
95999  96004  96009  96014  96019 
96047  96052  96057  96061  96066 
96095  96099  96104  96109  96114 

95928  95933  95938  95942  95947 
95976  95980  95985   95990  95995 
96023  96028  96033  96038  96042 
96071  96076  96080  96085  96090 
96118  96123  96128  96133  96137 

915 
916 
917 
918 
919 

96142  96147  96152  96156  96161 
96190  96194  96199  96204  96209 
96237  96242  96246  96251  96256 
96284  96289  96294  96298  96303 
96332  96336  96341  96346  96350 

96166  96171  96175  96180  96185 
96213  96218  96223  96227  96232 
96261  96265  96270  96275  96280 
96308  96313  96317  96322  96327 
96355  96360  96365  96369  96374 

920 

921 
922 
923 
924 

96379  96384  96388  96393  96398 
96426  96431  96435  96440  96445 
96473  96478  96483  96487  96492 
96520  96525  96530  96534  96539 
96567  96572  96577  96581   96586 

96402  96407  96412  96417  96421 
96450  96454  96459  96464  96468 
96497  96501  96506  96511  96515 
96544  96548  96553  96558  96562 
96591  96595  96600  96605  96609 

925 
926 
927 
928 
929 

96614  96619  96624  96628  96633 
96661  96666  96670  96675  96680 
96708  96713  96717  96722  96727 
96755  96759  96764  %  769  96774 
96802  96806  96811   96816  96820 

96638  96642  96647  96652  96656 
96685  96689  96694  96699  96703 
96731  96736  96741  96745  96750 
96778  96783  96788  96792  96797 
96825  96830  96834  96839  96844 

930 

931 
932 
933 
934 

96848  96853  96858  96862  96867 
96895  96900  96904  96909  96914 
96942  96946  96951  96956  96960 
96988  96993  96997  97002  97007 
97035  97039  97044  97049  97053 

96872  96876  96881   96886  96890 
96918  96923   96928  96932  9w937 
96965  96970  96974  96979  96984 
97011  97016  97021  97025  97030 
97058  97063  97067  97072  97077 

935 
936 
937 
938 
939 

97081  97086  97090  97095  97100 
97128  97132  97137  97142  97146 
97174  97179  97183  97188  97192 
97  220  97  225  .97  230  97  234  97  239 
97267  97,271  97276  97280  97285 

97104  97109  97114  97118  97123 
97151  97155  97160  97165  97169 
97197  97202  97206  97211   97216 
97243  97248  97253  97257  97262 
97290  97294  97299  97304  97308 

940 

941 
942 
943 

944 

97313  97317  97322  97327  97331 
97359  97364  97368  97373  97377 
97405  97410  97414  97419  97424 
97451  97456  97460  97465  97470 
97497  97502  97506  97511  97516 

97336  97340  97345  97350  97354 
97382  97387  97391  97396  97400 
97428  97433  97437  97442  97  M7 
97474  97479  97483  97488  97493 
97520  97525  97529  97534  97539 

945 
946 
947 
948 
949 

97543  97548  97552  97557  97562 
97589  97594  97598  97603  97607 
97635  97640  97644  97649  97653 
97681  97685  97690  97695  97699 
97727  97731  97736  97740  97745 

97566  97571  97575  97580  97585 
97612  97617  97621  97626  97630 
97658  97663  97667  97672  97676  , 
97704  97708  97713  97717  97722  ; 
97  749  97  754  97  759  97  763  97  768  , 

950 

97772  97777  97782  97786  97791 

97795  97800  97804  97809  97813 

N 

01234 

56789 

900-950 


950-1000 


19 


N 

950 

951 
952 
953 

954 

955 
956 
957 


O 


97772  97777  97782  97786  97791 
97818  97823  97827  97832  97836 
97864  97868  97873  97877  97882 
97909  97914  97918  97923  97928 
97955  97959  97964  97968  97973 

98000  98005  98009  98014  98019 
98046  98050  98055  98059  98064 
98091  98096  98100  98105  98109 
98137  98141  98146  98150  98155 
98182  98186  98191  98195  98200 

98227  98232  98236  98241  98245 
98272  98277  98  281  98286  98290 
98318  98322  98327  98331  98336 
98363  98367  98372  98376  98381 
98408  98412  98417  98421  98426 

98453  98457  98462  98466  98471 

98498  98502  98507  98511  98516 

98543  98547  98552  98556  98561 

98588  98592  98597  98601  98605 

98632  98637  98641  98646  98650 

98677  98682  98686  98691  98695 
98722  98726  98731  98735  98740 
98767  98771  98776  98780  98784 
98811  98816  98820  98825  98829 
98856  98860  98865  98869  98874 

98900  98905.  98909  98914  98918 

98  945.   98949  98954  98958  98  963 

98989  98994  98998  99003  99007 

99034  99038  99043  99047  99052 

99078  99083  99087  99092  99096 

99123  99127  99131   99136  99140 

99167  99171  99176  99180  99185 

99211  99216  99220  99224  99229 

99255  99260  99264  99269   99273 

99300  99304  99308  99313  99317 

99344  99348  99352  99357  99361 

99388  99392  99396  99401  99405 

99432  99436  99441  99445  99449 

99476  99480  99484  99489  99493 

99520  99524  99528  99533  99537 

99564  99568  99572  99577  99581 
99607  99612  99616  99621  99625 
99651  99656  99660  99664  99669 
99695  99699  99704  99708  99712 
99739  99743  99747  99752  99756 


99782  99787  99791 

99826  99830  99835 

99870  99874  99878 

99913  99917  99922 

99957  99961  99  965 

00000  00004  00009 


99795 
99839 
99883 
99926 
99970 

00013  00017 


99800 
99843 
99887 
99930 
99974 


8 


9 


97795  97800  97804  97809  97813 
97841  97845  97850  97855  97859 
97886  97891  97896  97900  97905 
97932  97937  97941  97946  97950 
97978  97982  97987  97991  979% 
98023  98028  98032  98037  98041 
98068  98073  98078  98082  98087 
98114  98118  98123  98127  98132 
98159  98164  98168  98173  98177 
98204  98209  98214  98218  98223 
98250  98254  98259  98263  98268 
98295  98299  98304  98308  98  3 13 
98340  98345  98349  98354  98358 
98385  98390  98394  98399  98403 
98430  98435  98439  98444  98  448 

98475  98480  98484  98489  98493 

98520  98525   98529  98534  98538 

98565  98570  98574  98579  98583 

98610  98614  98619   98623  98628 

98655  98659  98664  98668  98673 

98700  98704  98709  98713  98717 

98744  98749  98753  98758  98762 

98789  98793  98798  98802  98807 

98834  98838  98843  98847  98851 

98878  98883  98887  98892  988% 

98923  98927  98932  98936  98941 

98967   98972   98976  98981  98985 

99012   99016  99021  99025  99029 

99056  99061   99065  99069  99074 

99100  99105  99109  99114  99118 

99145  99149  99154  99158  99162 

99189  99193  99198  99202  99207 

99233  99238  99242  99247  99251 

99277  99282  99  286  99291   99295 

99322  99326  99330  99335  99339 

99366  99370  99374  99379  99383 

99410  99414   99419  99423  99427 

99454  99458  99463  99467  99471 

99498  99502  99506  99511  99515 

99542  99546  99550  99555  99559 

99585  99590  99594  99599  99603 

99629  99634  99638  99642  99647 

99673  99677  99682  99686  99691 

99717  99721   99726  99730  99734 

99760  99765  99769  99774  99778 


99813 
99  856 
99900 
99944 
99987 

00022  00026  00030 


99804 
99848 
99891 
99935 
99978 


99808 
99852 
998% 
99939 
99983 


99817 
99861 
99904 
99948 
99991 

00035 


99822 
99865 
99909 
99952 
999% 

00039 


0 


950-1000 


TABLE   II, 


APPROXIMATE  EQUATION  OF  TIME. 


DATE. 

MINUTES. 

DATE. 

MINUTES. 

DATE. 

MINUTES. 

DATE. 

MINUTES. 

Jan.     1 

4 

Apr.     1 

4   jj 

Aug.    9 

5  : 

Oct.    27 

16  ; 

"       3 

5 

4 

3  1 

"      15 

4   ^ 

Nov.  15 

15    • 

"       5 

6 

"       7 

2  ^ 

"      20 

3  | 

"      20 

14    I 

"       7 

7 

"     11 

1  J 

"      24 

2  | 

"      24 

13    ! 

«       9 

8 

"     15 

o  3 

"     28 

i  ; 

"      27 

12   g 

"     12 

9 

. 

"     31 

0    • 

"     30 

11  M 

"     15 

10 

"     19 

i  ; 

Dec.     2 

10  | 

"     18 

11    § 

"     24 

2   g 

Sept.    3 

i  ! 

"       5 

9  ^ 

"      21 

12  * 

"     30 

3| 

((           O 

2  : 

<«       7 

8   | 

-     25 

13  | 

May  13 

4  M 

«       9 

3  ; 

"       9 

7  55 

"     31 

14  ~ 

"      29 

3  | 

"      12 

4  ; 

"     11 

63 

Feb.  10 

15  | 

June    5 

2  0 

"     15 

5    ' 

"     13 

5  J 

"     21 

14  <S 

"     10 

i  ; 

"     18 

6  1 

"     16 

4  ^ 

"     27 

13  -g 

"     15 

o  • 

"     21 

7  1 

"      18 

3    ' 

Mar.    4 

125 

"      24 

8  44 

«      20 

2    ! 

"       8 

11 

"     20 

i  i 

"     27 

9   o 

«      22 

i  ; 

"     12 

10 

"     25 

2| 

"     30 

10" 

"      24 

o  • 

"      15 

9 

"     29 

3I 

Oct.     3 

11   • 

, 

"      19 

8 

July    5 

4  ^ 

6 

12  : 

«      26 

1  I 

"     22 

7 

"      11 

5  J 

"     10 

13  ; 

««     28 

2   * 

«     25 

6 

«     28 

6  ^ 

«     14 

14  ; 

««     30 

sl 

"     28 

5 

• 

"      19 

15    • 

1 

TABLE  III, 


THE  LOOA.KITHMS 

OF   THE 

TRIGONOMETRIC    FUNCTIONS 

Prom  0°  to  0°  3',  or  89°  57'  to  90°,  for  every  second; 
Prom  0°  to  2°,  or  88°  to  90°,  for  every  ten  seconds; 
Prom  1°  to  89°,  for  every  minute. 


NOTE.  To 

log  sin 

all  the  logarithms  —10  is  to  be  appended. 

0° 

log  tan  =  log  sk 
log  cos  =10.  00  000 

it 

0' 

1' 

2' 

t  r 

tt 

0' 

1' 

2' 

tt 

o 



6.  46  373 

6.  76  476 

60 

30 

6.  16  270 

6.  63  982 

6.  86  167 

30 

1 

4.  68  557 

6.47090 

6.  76  836 

59 

31 

6.  17  694 

6.  64  462 

6.  86  455 

29 

2 

4.98660 

6.  47  797 

6.  77  193 

58 

32 

6.  19  072 

6.  64  936 

6.  86  742 

28 

3 

5.  16  270 

6.  48  492 

6.  77  548 

57 

33 

6.  20  409 

6.  65  406 

6.  87  027 

27 

4 

5.  28  763 

6.49175 

6.77900 

56 

34 

6.  21  705 

6.  65  870 

6.  87  310 

26 

5 

5.  38  454 

6.49849 

6.  78  248 

55 

35 

6.  22  964 

6.  66  330 

6.  87  591 

25 

6 

5.  46  373 

6.  50  512 

6.  78  595 

54 

36 

6.  24  188 

6.  66  785 

6.  87  870 

24 

7 

5.  53  067 

6.  51  165 

6.  78  938 

53 

37 

6.  25  378 

6.  67  235 

6.  88  147 

23 

8 

5.  58  866 

6.  51  808 

6.  79  278 

52 

38 

6.  26  536 

6.  67  680 

6.  88  423 

22 

9 

5.  63  982 

6.  52  442 

6.  79  616 

51 

39 

6.27664 

6.  68  121 

6.  88  697 

21 

10 

5.  68  557 

6.  53  067 

6.  79  952 

50 

40 

6.  28  763 

6.  68  557 

6.  88  969 

20 

11 

5.  72  697 

6.  53  683 

6.  80  285 

49 

41 

6.  29  836 

6.  68  990 

6.  89  240 

19 

12 

5.  76  476 

6.  54  291 

6.  80  615 

48 

42 

6.  30  882 

6.  69  418 

6.  89  509 

18 

13 

5.  79  952 

6.  54  890 

6.  80  943 

47 

43 

6.31904 

6.  69  841 

6.  89  776 

17 

14 

5.  83  170 

6.  55  481 

6.  81  268 

46 

44 

6.  32  903 

6.  70  261 

6.90042 

16 

15 

5.  86  167 

6.  56  064 

6.  81  591 

45 

45 

6.  33  879 

6.  70  676 

6.90306 

15 

16 

5.  88  969 

6.  56  639 

6.  81  911 

44 

46 

6.  34  833 

6.  71  088 

6.90568 

14 

17 

5.  91  602 

6.  57  207 

6.  82  230 

43 

47 

6.  35  767 

6.  71  496 

6.  90  829 

13 

18 

5.  94  085 

6.  57  767 

6.  82  545 

42 

48 

6.36682 

6.71900 

6.  91  088 

12 

19 

5.  96  433 

6.  58  320 

6.  82  859 

41 

49 

6.  37  577 

6.  72  300 

6.  91  346 

11 

20 

5.98660 

6.  58  866 

6.  83  170 

40 

50 

6.38454 

6.  72  697 

6.  91  602 

10 

21 

6.  00  779 

6.  59  406 

6.  83  479 

39 

51 

6.39315 

6.73090 

6.91857 

9 

22 

6.  02  800 

6.  59  939 

6.  83  786 

38 

52 

6.  40  158 

6.  73  479 

6.  92  110 

8 

23 

6.  04  730 

6.  60  465 

6.  84  091 

37 

53 

6.  40  985 

6.  73  865 

6.  92  362 

7 

24 

6.  06  579 

6.  60  985 

6.84394 

36 

54 

6.  41  797 

6.  74  248 

6.  92  612 

6 

25 

6.  08  351 

6.  61  499 

6.84694 

35 

55 

6.42594 

6.  74  627 

6.  92  861 

5 

26 

6.  10  055 

6.  62  007 

6.  84  993 

34 

56 

6.  43  376 

6.  75  003 

6.  93  109 

4 

27 

6.  11  694 

6.  62  509 

6.  85  289 

33 

57 

6.44145 

6.  75  376 

6.  93  355 

3 

28 

6.  13  273 

6.  63  006 

6.  85  584 

32 

58 

6.44900 

6.  75  746 

6.93599 

2 

29 

6.  14  797 

6.  63  496 

6.  85  876 

31 

59 

6.  45  643 

6.  76  112 

6.93843 

1 

30 

6.  16  270 

6.  63  982 

6.  86  167 

30 

60 

6.  46  373 

6.  76  476 

6.94085 

0 

// 

59' 

58' 

57' 

tt 

tt 

59' 

58' 

57' 

f» 

log  cot  =  log  cos 
log  sin  =  10,  00  000 


log  cos 


22 


t  ft 

log  sin 

log  tan 

log  cos 

tf  t 

f  tt 

log  sin 

log  tan 

log  cos 

1  1   t 

0    0 

_ 



10.00000 

0  60 

1O  0 

7.  46  373 

7.  46  373 

10.00000 

0  50 

10 

5.  68  557 

5.68557 

10.00000 

50 

10 

7.  47  090 

7.  47  091 

10.00000 

50 

20 

5.98660 

5.98660 

10.00000 

40 

20 

7.  47  797 

7.  47  797 

10.00000 

40 

30 

6.  16  270 

6.  16  270 

10.00000 

30 

30 

7.  48  491 

7.  48  492 

10.00000 

30 

40 

6.  28  763 

6.  28  763 

10.00000 

20 

40 

7.49175 

7.  49  176 

10.00000 

20 

50 

6.38454 

6.  38  454 

10.00000 

10 

50 

7.  49  849 

7.  49  849 

10,00000 

10 

1    0 

6.46373 

6.  46  373 

10.00000 

0  59 

11   0 

7.50512 

7.50512 

10.00000 

0  49 

10 

6.  53  067 

6.  53  067 

10.00000 

50 

10 

7.  51  165 

7.  51  165 

10.00000 

50 

20 

6.58866 

6.  58  866 

10.00000 

40 

20 

7.  51  808 

7.  51  809 

10.00000 

40 

30 

6.  63  982 

6.  63  982 

10.00000 

30 

30 

7.  52  442 

7.  52  443 

10.00000 

30 

40 

6.  68  557 

6.  68  557 

10.00000 

20 

40 

7.  53  067 

7.  53  067 

10.00000 

20 

50 

6.  72  697 

6.  72  697 

10.00000 

10 

50 

7.  53  683 

7.  53  683 

10.00000 

10 

2    0 

6.76476 

6.76476 

10.00000 

0  58 

12  0 

7.  54  291 

7.  54  291 

10.00000 

0  48 

10 

6.  79  952 

6.79952 

10.00000 

50 

10 

7.  54  890 

7.  54  890 

10.00000 

50 

20 

6.  83  170 

6.  83  170 

10.00000 

40 

20 

7.  55  481 

7.  55  481 

10.00000 

40 

30 

6.  86  167 

6.  86  167 

10.00000 

30 

30 

7.  56  064 

7.56064 

10.00000 

30 

40 

6.88969 

6.  88  969 

10.00000 

20 

40 

7.  56  639 

7.  56  639 

10.00000 

20 

50 

6.  91  602 

6.  91  602 

10.00000 

10 

50 

7.  57  206 

7.  57  207 

10.00000 

10 

3    0 

6.  94  085 

6.94085 

10.00000 

0  57 

13  0 

7.  57  767 

7.  57  767 

10.00000 

0  47 

10 

6.96433 

6.96433 

10.00000 

50 

10 

7.58320 

7.  58  320 

10.00000 

50 

20 

6.98660 

6.  98  661 

10.00000 

40 

20 

7.58866 

7.  58  867 

10.00000 

40 

30 

7.00779 

7.  00  779 

10.00000 

30 

30 

7.59406 

7.59406 

10.00000 

30 

40 

7.  02  800 

7.  02  800 

10.00000 

20 

40 

7.59939 

7.59939 

10.00000 

20 

50 

7.  04  730 

7.  04  730 

10.00000 

10 

50 

7.60465 

7.  60  466 

10.00000 

10 

4    0 

7.  06  579 

7.06579 

10.00000 

056 

14  0 

7.60985 

7.  60  986 

10.00000 

0  46 

10 

7.08351 

7.08352 

10.00000 

50 

10 

7.  61  499 

7.  61  500 

10.00000 

50 

20 

7.10055 

7.10055 

10.00000 

40 

20 

7.  62  007 

7.62008 

10.00000 

40 

30 

7.  11  694 

7.  11  694 

10.00000 

30 

30 

7.  62  509 

7.  62  510 

10.00000 

30 

40 

7.  13  273 

7.  13  273 

10.00000 

20 

40 

7.63006 

7.  63  006 

10.00000 

20 

50 

7.  14  797 

7.  14  797 

10.00000 

10 

50 

7.  63  496 

7.  63  497 

10.00000 

10 

5    0 

7.  16  270 

7.16270 

10.00000 

0  55 

15  0 

7.  63  982 

7.  63  982 

10.00000 

0  45 

10 

7.17694 

7.  17  694 

10.00000 

50 

10 

7.  64  461 

7.64462 

10.00000 

50 

20 

7.19072 

7.  19  073 

10.00000 

40 

20 

7.  64  936 

7.64937 

10.00000 

40 

30 

7.20409 

7.20409 

10.00000 

30 

30 

7.  65  406 

7.  65  406 

10.00000 

30 

40 

7.  21  705 

7.  21  705 

10.00000 

20 

40 

7.  65  870 

7.  65  871 

10.00000 

20 

50 

7.  22  964 

7.  22  964 

10.00000 

10 

50 

7.  66  330 

7.  66  330 

10.00000 

10 

6    0 

7.  24  188 

7.  24  188 

10.00000 

0  54 

16  0 

7.  66  784 

7.  66  785 

10.00000 

0  44 

10 

7.  25  378 

7.  25  378 

10.00000 

50 

10 

7.  67  235 

7.  67  235. 

10.00000 

50 

20 

7.  26  536 

7.  26  536 

10.00000 

40 

20 

7.  67  680 

7.  67  680 

10.00000 

40 

30 

7.  27  664 

7.27664 

10.00000 

30 

30 

7.  68  121 

7.  68  121 

10.00000 

30 

40 

7.  28  763 

7.  28  764 

10.00000 

20 

40 

7.68557 

7.  68  558 

9.99999 

20 

50 

7.  29  836 

7.  29  836 

10.00000 

10 

50 

7.  68  989 

7.68990 

9.99999 

10 

7    0 

7.30882 

7.30882 

10.00000 

0  53 

17  0 

7.69417 

7.  69  418 

9.99999 

0  43 

10 

7.31904 

7.31904 

10.00000 

50 

10 

7.  69  841 

7.  69  842 

9.99999 

50 

20 

7.  32  903 

7.  32  903 

10.00000 

40 

20 

7.  70  261 

7.  70  261 

9.99999 

40 

30 

7.  33  879 

7.  33  879 

10.00000 

30 

30 

7.  70  676 

7.  70  677 

9.99999 

30 

40 

7.  34  833 

7.  34  833 

10.00000 

20 

40 

7.  71  088 

7.  71  088 

9.99999 

20 

50 

7.  35  767 

7.  35  767 

10.00000 

10 

50 

7.  71  496 

7.  71  496 

9.99999 

10 

8    0 

7.  36  682 

7.36682 

10.00000 

0  52 

18  0 

7.  71  900 

7.71900 

9.99999 

0  42 

10 

7.37577 

7.37577 

10.00000 

50 

10 

7.  72  300 

7.72301 

9.99999 

50 

20 

7.38454 

7.38455 

10.00000 

40 

20 

7.  72  697 

7.72697 

9.99999 

40 

30 

7.39314 

7.39315 

10.00000 

30 

30 

7.73090 

7.73090 

9.99999 

30 

40 

7.40158 

7.  40  158 

10.00000 

20 

40 

7.  73  479 

7.  73  480 

9.99999 

20 

50 

7.  40  985 

7.  40  985 

10.00000 

10 

50 

7.  73  865 

7.  73  866 

9.99999 

10 

9    0 

7.  41  797 

7.  41  797 

10.00000 

0  51 

19  0 

7.  74  248 

7.  74  248 

9.99999 

0  41 

10 

7.  42  594 

7.  42  594 

10.00000 

50 

10 

7.  74  627 

7.  74  628 

9.99999 

50 

20 

7.  43  376 

7.  43  376 

10.00000 

40 

20 

7.  75  003 

7.75004 

9.99999 

40 

30 

7.  44  145 

7.  44  145 

10.00000 

30 

30 

7.  75  376 

7.  75  377 

9.99999 

30 

40 

7.  44  900 

7.44900 

10.00000 

20 

40 

7.  75  745 

7.  75  746 

9.99999 

20 

50 

7.  45  643 

7.  45  643 

10.00000 

10 

50 

7.  76  112 

7.  76  113 

9.99999 

10 

100 

7.46373 

7.  46  373 

10.00000 

0  50 

20  0 

7.76475 

7.76476 

9.99999 

0  40 

t  ft 

log  cos 

log  cot 

log  sin 

tt  f 

t   tt 

log  cos 

log  cot 

log  sin 

r  t   t 

89' 


r  tt 

log  sin. 

log  tan 

log  cos 

ft   t 

f  tf 

log  sin 

log  tan 

log  cos 

ft  t 

20  0 

7.76475 

7.76476 

9.99999 

0  40 

300 

7.94084 

7.  94  086 

9.99998 

0  3O 

10 

7.  76  836 

7.76837 

9.99999 

50 

10 

7.  94  32^ 

7.  94  326 

9.99998 

50 

20 

7.  77  193 

7.  77  194 

9.99999 

40 

20 

7.94564 

7.94566 

9.99998 

40 

30 

7.  77  548 

7.  77  549 

9.99999 

30 

30 

7.94802 

7.94804 

9.99998 

30 

40 

7.  77  899 

7.77900 

9.99999 

20 

40 

7.  95  039 

7.95040 

9.99998 

20 

50 

7.  78  248 

7.  78  249 

9.99999 

10 

50 

7.  95  274 

7.  95  276 

9.99998 

10 

21  0 

7.  78  594 

7.  78  595 

9.99999 

0  39 

31  0 

7.  95  508 

7.  95  510 

9.99998 

0  29 

10 

7.  78  938 

7.  78  938 

9.99999 

50 

10 

7.  95  741 

7.  95  743 

9.99998 

50 

20 

7.  79  278 

7.  79  279 

9.99999 

40 

20 

7.  95  973 

7.  95  974 

9.99998 

40 

30 

7.  79  616 

7.79617 

9.99999 

30 

30 

7.  96  203 

7.  96  205 

9.99998 

30 

40 

7.  79  952 

7.  79  952 

9.99999 

20 

40 

7.96432 

7.%  434 

9.99998 

20 

50 

7.  80  284 

7.  80  285 

9.99999 

10 

50 

7.96660 

7.96662 

9.99998 

10 

22  0 

7.80615 

7.  80  615 

9.99999 

0  38 

32  0 

7.  96  887 

7.96889 

9.99998 

0  28 

10 

7.  80  942 

7.  80  943 

9.  99  999 

50 

10 

7.97113 

7.97114 

9.99998 

50 

20 

7.  81  268 

7.  81  269 

9.99999 

40 

20 

7.  97  337 

7.  97  339 

9.99998 

40 

30 

7.81591 

7.  81  591 

9.99999 

30 

30 

7.  97  560 

7.  97  562 

9.99998 

30 

40 

7.81911 

7.81912 

9.  99  999 

20 

40 

7.  97  782 

7.  97  784 

9.99998 

20 

50 

7.  82  229 

7.  82  230 

9.99999 

10 

50 

7.  98  003 

7.98005 

9.99998 

10 

23  0 

7.  82  545 

7.  82  546 

9.99999 

0  37 

33  0 

7.  98  223 

7.  98  225 

9.99998 

0  27 

10 

7.  82  859 

7.  82  860 

9.99999 

50 

10 

7.  98  442 

7.98444 

9.99998 

50 

20 

7.  83  170 

7.  83  171 

9.99999 

40 

20 

7.98660 

7.98662 

9.99998 

40 

30 

7.  83  479 

7.  83  480 

9.99999 

30 

30 

7.  98  876 

7.98878 

9.99998 

30 

40 

7.  83  786 

7.  83  787 

9.99999 

20 

40 

7.99092 

7.99094 

9.99998 

20 

50 

7.  84  091 

7.  84  092 

9.99999 

10 

50 

7.99306 

7.99308 

9.99998 

10 

24  0 

7.84393 

7.  84  394 

9.99999 

0  36 

34  0 

7.  99  520 

7.  99  522 

9.99998 

0  26 

10 

7.  84  694 

7.  84  695 

9.99999 

50 

10 

7.  99  732 

7.  99  734 

9.99998 

50 

20 

7.  84  992 

7.84994 

9.  99  999 

40 

20 

7.  99  943 

7.  99  946 

9.99998 

40 

30 

7.  85  289 

7.  85  290 

9.99999 

30 

30 

8.  00  154 

8.  00  156 

9.99998 

30 

40 

7.  85  583 

7.  85  584 

9.99999 

20 

40 

8.00363 

8.00365 

9.99998 

20 

50 

7.  85  876 

7.85877 

9.99999 

10 

50 

8.00571 

8.  00  574 

9.99998 

10 

25  0 

7.  86  166 

7.  86  167 

9.99999 

0  35 

35  0 

8.  00  779 

8.  00  781 

9.99998 

0  25 

10 

7.86455 

7.  86  456 

9.99999 

50 

10 

8.00985. 

8.  00  987 

9.99998 

50 

20 

7.  86  741 

7.  86  743 

9.  99  999 

40 

20 

8.  01  190 

8.  01  193 

9.99998 

40 

30 

7.  87  026 

7.  87  027 

9.  99  999 

30 

30 

8.  01  395 

8.  01  397 

9.99998 

30 

40 

7.  87  309 

7.87310 

9.  99  999 

20 

40 

8.  01  598 

8.01600 

9.99998 

20 

50 

7.  87  590 

7.  87  591 

9.99999 

10 

50 

8.  01  801 

8.  01  803 

9.99998 

10 

26  0 

7.  87  870 

7.  87  871 

9.99999 

0  34 

36  0 

8.  02  002 

8.02004 

9.99998 

0  24 

10 

7.  88  147 

7.  88  148 

9.99999 

50 

10 

8.  02  203 

8.  02  205 

9.99998 

50 

20 

7.  88  423 

7.  88  424 

9.99999 

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8.  02  402 

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27  0 

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9.99999 

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8.  03  192 

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7.90305 

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7.  90  829 

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9.99999 

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8.  04  159 

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28  0 

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9.99997 

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8.04540 

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8.04918 

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47 

84  784 

99672 

00328 

85  112 

13 

48 

84796 

99697 

00303 

85  100 

12 

49 

84  809 

99722 

00278 

85087 

11 

5O 

84822 

99747 

00253 

85074 

10 

51 

84835 

99773 

00227 

85062 

9 

52 

84847 

99798 

00202 

85049 

8 

63 

84860 

99823 

00177 

85037 

7 

54 

84873 

99848 

00152 

85024 

6 

55 

84  885 

99874 

00  126 

85  012 

5 

56 

84898 

99899 

00101 

84999 

4 

57 

84911 

99924 

00076 

84986 

3 

58 

84923 

99949 

00051 

84974 

2 

59 

84936 

W  975 

00025 

84961 

1 

60 

84949 

00  OCX) 

00000 

84949 

0 

10 

10 

g 

t 

log  cos 

log  cot 

log  tan 

log  sin 

» 

46 


45 


50 


TABLE  IV. 


FOB  DETERMINING  WITH  GREATER  ACCURACY  THAN  CAN  BE  DONE  BY 
MEANS  OF    TABLE  III. : 

1.  log  sin,  log  tan,  and  log  cot,  when  the  angle  is  between  0°  and  2°  ; 

2.  log  cos,  log  tan,  and  log  cot,  when  the  angle  is  between  88°  and  90°  ; 

3.  The  value  of  the  angle  when  the  logarithm  of  the  function  does  not 

lie  between  the  limits  8.  54  684  and  11.  45  316. 


FORMULAS  FOR  THE  USE   OF  THE  NUMBERS   S  AND  T. 
I.   When  the  angle  a  is  between  0°  and  2°  : 


log  sin  a  =  log  a"  +  S. 
log  tan  a  =  log  a"  +  T. 
log  cot  a  =  colog  tan  a. 


log  a"  =  log  sin  a  —  S, 
=  log  tana—  T, 
=  colog  cot  a  —  T. 


II.   When  the  angle  a  is  between  88°  and  90°  : 

log  COS  a  =  log  (90" -a)"  +  S. 
log  COt  a  =  log  (90°-a)"+  r. 

log  tan  a  =  colog  cot  a. 


log  (90°— a)"  =  log  cos  a— S, 
=  log  COt  a—  T, 

-  colog  tana-r, 
and  a  =  90° -(90° -a). 


TALIIES  OF  S  AISTD  T, 


a" 

3 

log  sin  a 

a" 

T 

log  tan  a 

a 

T 

log  tan  a 

0 

_ 

0 

_ 

5146 

8.  39  713 

4.68557 

4.68557 

4.  68  567 

2409 

8.06740 

200 

6,98660 

5424 

8.41999 

4.68556 

4.  68  558 

4.  68  568 

3417 

8.  21  920 

1726 

7.  92  263 

5689 

8.  44  072 

4.  68  555 

4.  68  559 

4.  68  569 

3823 

8.  26  795 

2432 

8.  07  156 

5941 

S.  45  955 

4.68555 

4.68560 

4.  68  570 

4190 

8.30776 

2976 

8.  15  924 

6184 

8.  47  697 

4.  68  554 

4.  68  561 

4.  68  571 

4840 

8.  37  038 

3434 

8.  22  142 

6417 

8.  49  305 

4.  68  553 

4.  68  562 

4.  68  572 

5414 

8.41904 

3838 

S.  26  973 

6642 

8.50802 

4.  68  552 

4.  68  563 

4.  68  573 

5932 

8.  45  872 

4204 

8.  30  930 

6859 

8.  52  200 

4.  68  551 

4.68564 

4.  68  574 

6408 

8.  49  223 

4540 

8.  34  270 

7070 

S.  53516 

4.  68  550 

4.  68  565 

4.68575 

6633 

8.  50  721 

4699 

8.  35  766  ! 

7173 

8.  54  145 

4.  68  550 

4.  68  565 

4.  68  575 

6851 

8.  52  125 

4853 

8.  37  167 

7274 

8.  54  753 

4.  68  549 

4.68566 

7267 

8.54684 

5146 

8.  39  713 

a." 

S 

log  sin  a 

a" 

T 

log  tan  a 

a 

T 

log  tan  a 

51 


TABLE   IV. 

This  table  (page  50)  must  be  used  when  great  accuracy  is  desired  in 
working  with  angles  between  0°  and  2°,  or  between  88°  and  90°. 

The  values  of  S  and  T  are  such  that  when  the  angle  a  is  expressed 
in  seconds, 

8  =  log  sin  a  —  log  a", 

T  =  log  tan  a  —  log  a". 

Hence,  follow  the  formulas  given  on  the  page  containing  the  table. 
The  values  of  S  and  T  are  printed  with  the  characteristic  10  too 
large,  and  in  using  them  —10  must  always  be  annexed. 


Find  log  sin  0°  58'  17". 
0°  58'  17"  =  3497." 
log  3497  =  3.54370 

8  =  4.68555-10 


log  sin  0°  58'  17"  =  8.22925  - 10 

Find  log  tan  0°  52'  47.5". 

0°  52'  47.5"  =  3167.5." 

log  3167.5  =  3.50072 

T  =  4.68561 -10 

log  tan  0°  52'  47.5"  =  8.18633-10 


Find  log  cos  88°  26' 41.2". 
90°  -  88°  26'  41.2"  =  1°  33'  18.8* 


log  5598.8  =  3.74809 

8  =  4.68552-10 

log  cos  88°  26'  41.2"  =  8.43361-10 

Find  log  tan  89°  54'  37.362". 
90°  -  89°  54'  37.362"  =  322.638*. 
log  322.638  =  2.60871 

T=  5.68558 -10 

log  cot  89' 54' 37 .362"=  7.19429-10 
log  tan  89°  54  37.362 ''  =  2.80671. 


Find  the  angle,  if  log  sin  =  6.72306  -  10. 

6.72306  - 10 
8  =  4.68557-10 

Subtract,     2.03749  =  log  10^.015. 

109.015"  =  0>  1   49.016''. 

Find  the  angle  for  which  log  cot  =  1 .67604. 
colog  cot  =  8.32396  -  10 
T  =  4.68564  -  10 

Subtract,      3.63832  =  log  4348.3. 

4348.3"  =  1°  12'  28.3". 

Find  the  angle  for  which  log  tan  =  1.55407. 
colog  tan  =  8.44593  —  10 
T  =  4.68569  -  10 

Subtract,      3.76024          =  log  6757.6. 

5757.6"  =  1°  35'  57.6", 
and      90°  - 1°  35'  67.6"  =  88°  24'  2.4"  =  angle  required 


52 


TABLE  V, 


SHOWING  LENGTHS  IN  NAUTICAL  MILES  AND  STATUTE  MILES  OF  DEGREES 
OF  LATITUDE  AND  LONGITUDE  IN  DIFFERENT  LATITUDES. 


DEGREE  OF  THE  PARALLEL. 

DEGREE  OF  THE  MERIDIAN. 

Latitude 
of 
Parallel. 

Nautical 
Miles. 

Statute 
Miles. 

Latitude 
of  Middle 
Point. 

Nautical 
Miles. 

Statute 
Miles. 

20° 

56.404 

65.018 

20° 

59.664 

68.777 

21° 

56.039 

64.598 

22° 

55.657 

64.158 

23° 

55.258 

63.698 

24° 

54.843 

63.219 

25° 

54.411 

62.721 

25° 

59.706 

68.825 

26° 

53.962 

62.204 

27° 

53.497 

61.668 

28° 

53.016 

61.113 

29° 

52.518 

60.540 

30° 

52.005 

59.948 

30° 

59.749 

68.875 

31° 

51.476 

59.338 

32° 

50.931 

58.709 

33° 

50.370 

58.063 

34° 

49.794 

57.399 

35° 

49.203 

56.718 

35° 

59.7% 

68.929 

36° 

48.597 

56.019 

37° 

47.976 

55.304 

38° 

47.341 

54.571 

39° 

46.960 

53.822 

40° 

46.026 

53.056 

40° 

59.847 

68.987 

41° 

45.348 

52.274 

42° 

44.654 

51.476 

43° 

43.949 

50.662 

44° 

43.230 

49.833 

45° 

42.497 

48.988 

45° 

59.899 

69.048 

46° 

41.752 

48.128 

47° 

40.993 

47.254 

48° 

40.222 

46.365 

490 

39.439 

45.462 

50° 

38.643 

44.545 

50° 

59.951 

69.108 

53 


TABLE   VI, 


MISCELLANEOUS  FORMULAE,  AND  EQUIVALENTS  OF  METRES,  CHAINS, 
AND  FEET. 


:  3.14159265 
:  0.31830989 
:  9.86960440 
:  0.10132118 


Logarithm. 
0.4971499 

9.5028501  — 10 

0.9942997 

9.0057006-10 


:  1.77245385 
0.56418958 

1.46459189 


Logarithm. 
0.2485749 
9.7514251  -  10 

0.1657166 


Circumference  of  circle,  diameter  being  unity, 

Area  of  circle,  radius  being  unity 

Surface  of  sphere,  diameter  being  unity  .  .  .  . 
Area  of  a  circle,  diameter  being  unity 


Volume  of  sphere,  diameter  being  unity 


Volume  of  sphere,  radius  being  unity .  . 
Arc  whose  length  is  equal  to  the  radius : 
Expressed  in  degrees  


Expressed  in  minutes    . 


Expressed  in  seconds    .  .  .  .  , 
If  radius  is  unity : 

Length  of  arc  for  one  degree 

Length  of  arc  for  one  minute 

Length  of  arc  for  one  second , 
Sine  of  one  second 


180 


10800 


648000 


IT 

180 


10800 


648000 


Base  of  Hyperbolic  or  Napier's  System  of  Logarithms  . : 
Modulus  of  Common  or  Briggs'  System  of  Logarithms  . : 

Equatorial  radius  of  the  earth  in  feet 

Polar  radius  of  the  earth  in  feet ; 

Length  of  degree  of  latitude  at  the  equator,  in  feet    .  . : 
Length  of  degree  of  latitude  at  46°,  in  feet : 


:  3.14159265 
:  0.7853982 
:  0.52359878 
:  4.1887902 

=  57.2957795° 
=  3437.74677' 
:  206264.806 

=  0.0174533 
=  0.0002909 

=  0.00000485 

=  0.00000485 
;  2.7182818 
0.4342945 
: 20923600 
20853657 
362748.33 
364571.77 


0.4971499 
9.8950899  - 10 
9.7189986-10 
0.6220886 

1.7581226 
3.5362739 
5.3144251 

8.2418774  - 10 
6.4637261  - 10 

4.68557487  - 10 

4.68557487  - 10 
0.4342945 
9.6377843-10 


FEET.     METRES.    CHAINS.     METRES 


0.3048 
0.6096 
0.9144 
1.2192 
1.52-10 
1.8288 
2.1336 
2.4384 
2.7432 
3.0480 


0.0151 
0.0303 
0.0455 
0.0606 
0.0758 
0.0909 
0.1061 
0.1212 
0.1364 
0.1515 


FEET. 


3.2809 

6.5617 

9.8426 

13.1235 

16.4044 

19.6852 

22.9661 

26.2470 

29.5278 

32.8087 


54 


TABLE  VII.  — TRAVERSE  TABLE.   . 

This  table  gives  the  latitude  and  departure  to  three  places  of  deci- 
mals for  distances  from  1  to  10,  corresponding  to  bearings  from 
0°  to  90°,  at  intervals  of  15'. 

If  the  bearing  does  not  exceed  45°,  it  is  found  in  the  left-hand 
column,  and  the  designations  of  the  columns  under  "Distance"  are 
taken  from  the  top  of  the  page  ;  but  if  the  bearing  exceeds  45°,  it  is 
found  in  the  right-hand  column,  and  the  designations  of  the  columns 
under  "Distance"  are  taken  from  the  bottom  of  the  page. 

The  method  of  using  the  table  will  be  made  plain  by  the  following 
examples : 

1.  Let  it  be  required  to  find  the  latitude  and  departure  of  a  line 
running  N.  35°  15'  E.  6  chains. 

On  page  60,  left-hand  column,  look  for  35°  15' ;  opposite  this  bearing,  in  the 
vertical  column  headed  "Distance  6,"  are  found  4.900  and  3.463,  under  the  head- 
ings "Latitude"  and  "Departure"  respectively.  Hence  latitude,  or  northing, 
=  4.900  chains,  and  departure,  or  easting,  =  3.463  chains. 

2.  Let  it  be  required  to  find  the  latitude  and  departure  of  a  Hue 
running  S.  87°  W.  2  chains. 

As  the  bearing  exceeds  45°,  we  look  in  the  right-hand  column  on  page  55,  and 
opposite  87°,  in  the  column  marked  "  Distance  2,"  we  find  (taking  the  designa- 
tions of  the  columns  from  the  bottom  of  the  page)  latitude  =  0.105  chains,  and 
departure  =  1.997  chains.  Hence  latitude,  or  southing,  =  0.105  chains,  and  depart- 
ure, or  westing,  =  1.997  chains. 

3.  Let  it  be  required  to  find  the  latitude  and  departure  of  a  line 
running  N.  15°  45'  W.  27.36  chains. 

In  this  case,  we  find  the  required  number  for  each  figure  of  the  distance  sepa- 
rately, arranging  the  work  as  in  the  following  table.  In  practice,  only  the  last 
columns  under  "  Latitude  "  and  "  Departure  "  are  written. 


Distance. 

Latitude. 

Departure. 

20      =  2  X  10 

7 
0.3   =3--  10 
0.06  =  6  -5-  100 

1.925x10   =19.25 
6.737 
2.887  +  10    =0.289 
5.775  -T-  100  =  0.058 

0.543  X    10  =  5.43 
1.90 
0.814+   10  =  0.081 
1.628  -T-  100  =  0.016 

27.36 

26.334 

7.427 

Hence  latitude  =  26.334  chains,  and  departure  =  7.427  chains. 


55 


A  TABLE   OF  THE  ANGLES 

Which  every  Point  and  Quarter  Point  of  the  Compass  makes  with  the  Meridian. 


North. 

Points. 

1 

o     1    II 
2  48  45 
5  37  30 
8  26  15 
11  15    0 

Points. 

0  —  V 
1 

South. 

N.  by  E. 

N.  by  W. 

S.  by  E. 

S.  by  W. 

N.N.E. 

N.N.W. 

2 

14    3  45 
16  52  30 
19  41  15 
22  30    0 

2 

S.S.E. 

S.S.W. 

N.E.  by  N. 

N.W.  by  N. 

2-!/4 

P 

25  18  45 
28    7  30 
30  56  15 
33  45    0 

2~$ 
3 

S.E.  by  S. 

S.W.  by  S. 

N.E. 

N.W. 

4      * 

36  33  45 
39  22  30 
42  11  15 
45    0    0 

3_i£ 
4 

S.E. 

S.W. 

N.E.byE. 

N.W.  by  W. 

l:fc 
J:< 

47  48  45 
50  37  80 
53  26  15 
56  15    0 

4—  8i 

5 

S.E.  by  E. 

S.W.  by  W. 

E.N.E. 

W.N.W. 

5-^ 
6 

If 

59    3  45 
61  62  30 
64  41  15 

C7  30    0 
70  IS  45 
73    7  30 
75  56  15 
78  45    0 

!:& 

!:< 

E.S.E. 

W.S.W. 

E.  by  N. 

W.  by  N. 

6-V* 
7     "* 

E.  by  S. 

W.  by  S. 

East. 

West. 

1:8 

T-% 

g 

81  33  46 
84  22  30 
87  11  15 
90    0    0 

J3 

East. 

West. 

56 


TABLE  VII.  -  TRAVERSE  TABLE. 


Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

o    r 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

0        / 

015 

1.000 

0.004 

2.000 

0.009 

3.000 

0.013 

4.000 

0.017 

5.000 

0.022 

8945 

30 

1.000 

0.009 

2.000 

0.017 

3.000 

0.026 

4.000 

0.035 

5.000 

0.044 

30 

45 

1.000 

0.013 

2.000 

0.026 

3.000 

0.039 

4.0CO 

0.052 

5.000 

0.065 

15 

1    0 

1.000 

0.017 

2.000 

0.035 

3.000 

0.052 

3.999 

0.070 

4.999 

0.087 

89    0 

15 

1.000 

0.022 

2.000 

0.044 

2.999 

0.065 

3.999 

O.OS7 

4.999 

0.109 

45 

30 

1.000 

0.026 

1.999 

0.052 

2.999 

0.079 

3.999 

0.105 

4.998 

0.131 

30 

45 

1.000 

0.031 

1.999 

0.061 

2.999 

0.092 

3.998 

0.122 

4.998 

0.153 

15 

2    0 

0.999 

0.035 

1.999 

0.070 

2.998 

0.105 

3.998 

0.140 

4.997 

0.174 

88    0 

15 

0.999 

0.039 

1.998 

0.079 

2.998 

0.118 

3.997 

0.157 

4.996 

0.196 

45 

30 

0.999 

0.044 

1.998 

0.087 

2.997 

0.131 

3.996 

0.174 

4.995 

0.218 

30 

45 

0.999 

0.048 

1.998 

0.096 

2.997 

0.144 

3.995 

0.192 

4.994 

0.240 

15 

3    0 

0.999 

0.052 

1.997 

0.105 

2.996 

0.157 

3.995 

0.209 

4.993 

0.262 

87    0 

15 

0.998 

0.057 

1.997 

0.113 

2.995 

0.170 

3.994 

0.227 

4.992 

0.2S3 

45 

30 

0.998 

0.061 

1.996 

0.122 

2.994 

0.183 

3.993 

0.244 

4.991 

0.305 

30 

45 

0.998 

0.065 

1.996 

0.131 

2.994 

0.196 

3.991 

0.262 

4.989 

0.327 

15 

4   0 

0.998 

0.070 

1.995 

0.140 

2.993 

0.209 

3.990 

0.279 

4.988 

0.349 

86    0 

15 

0.997 

0.074 

1.995 

0.148 

2.992 

0.222 

3.  989 

0.296 

4.986 

0.371 

45 

30 

0.997 

0.078 

1.994 

0.157 

2.991 

0.235 

3.988 

0.314 

4.985 

0.392 

30 

45 

0.997 

0.083 

1.993 

0.166 

2.990 

0.248 

3.986 

0.331 

4.983 

0.414 

15 

5   0 

0.996 

0.087 

1.992 

0.174 

2.989 

0.261 

3.985 

0.349 

4.9S1 

0.436 

85    0 

15 

0.996 

0.092 

1.992 

0,183 

2.987 

0.275 

3.983 

0.366 

4.979 

0.458 

45 

30 

0.995 

0.096 

1.991 

0.192 

2.986 

0.2SS 

3.982 

0.3S3 

4.977 

0.479 

30 

45 

0.995 

0.100 

1.990 

0.200 

2.9S5 

0.301 

3.980 

0.401 

4.975 

0.501 

15 

6   0 

0.995 

0.105 

1.989 

0.209 

2.984 

0.314 

3.978 

0.418 

4.973 

0.523 

84    0 

15 

0.994 

0.109 

1.988 

0.218 

2.9S2 

0.327 

3.976 

0.435 

4.970 

0.544 

45 

30 

0.994 

0.113 

1.9S7 

0.226 

2.981 

0.340 

3.974 

0.453 

4.968 

0566 

30 

45 

0.993 

0.118 

1.986 

0.235 

2.979 

0.353 

3.972 

0.470 

4.965 

0.588 

15 

7    0 

0.993 

0.122 

1.985 

0.244 

2.978 

0.366 

3.970 

0.4S7 

4.963 

0.609 

83    0 

15 

0.992 

0.126 

1.984 

0.252 

2.976 

0.379 

3.96S 

0.505 

4.960 

0.631 

45 

30 

0.991 

0.131 

1.983 

0.261 

2.974 

0.392 

3.966 

0.522 

4.957 

0.653 

30 

45 

0.991 

0.135 

1.982 

0.270 

2.973 

0.405 

3.963 

0.539 

4.954 

0.674 

15 

8   0 

0.990 

0.139 

1.981 

0.278 

2.971 

0.418 

3.961 

0.557 

4.951 

0.696 

82    0 

15 

0.990 

0.143 

1.979 

0.287 

2.969 

0.430 

3.959 

0.574 

4.948 

0.717 

45 

30 

0.989 

0.148 

1.978 

0.296 

2.967 

0.443 

3.956 

0.591 

4.945 

0.739 

30 

45 

0.988 

0.152 

1.977 

0.304 

2.965 

0.456 

3.953 

0.608 

4.942 

0.761 

15 

9    0 

0.988 

0.156 

1.975 

0.313 

2.963 

0.469 

3.951 

0.626 

4.938 

0.782 

81    0 

15 

0.987 

0.161 

1.974 

0.321 

2.961 

0.4S2 

3.948 

0.643 

4.935 

0.804 

45 

30 

0.986 

0.165 

1.973 

0.330 

2.959 

0.495 

3.945 

0.660 

4.931 

0.825 

30 

45 

0.986 

0.169 

1.971 

0.339 

2.957 

0.508 

3.942 

0.677 

4.928 

0.847 

15 

1O    0 

0.985 

0.174 

1.970 

0.347 

2.954 

0.521 

3.939 

0.695 

4.924 

0.868 

80    0 

15 

0.984 

0.178 

1.968 

0.356 

2.952 

0.534 

3.936 

0.712 

4.920 

0.890 

45 

30 

0.983 

0.182 

1.967 

0.364 

2.950 

0.547 

3.933 

0.729 

4.916 

0.911 

30 

45 

0.982 

0.187 

1.965 

0.373 

2.947 

0.560 

3.930 

0.746 

4.912 

0.933 

15 

11   0 

0.982 

0.191 

1.963 

0.382 

2.945 

0.572 

3.927 

0.763 

4.908 

0.954 

79    0 

15 

0.981 

0.195 

1.962 

0.390 

2.942 

0.585 

3.923 

0.780 

4.904 

0.975 

45 

30 

0.980 

0.199 

1.960 

0.399 

2.940 

0.598 

3.920 

0.797 

4.900 

0.997 

30 

45 

0.979 

0.204 

1.958 

0.407 

2.937 

0.611 

3.916 

0.815 

4.895 

1.018 

15 

12    0 

0.978 

0.208 

1.956 

0.416 

2.934 

0.624 

3.913 

0.832 

4.891 

1.040 

78    0 

15 

0.977 

0.212 

1.954 

0.424 

2.932 

0.637 

3.909 

0.849 

4.886 

1.061 

45 

30 

0.976 

0.216 

1.953 

0.433 

2.929 

0.649 

3.905 

0.866 

4.881 

1.082 

30 

45 

0.975 

0.221 

1.951 

0.441 

2.926 

0.662 

3.901 

O.S83 

4.877 

1.103 

15 

13    0 

0.974 

0.225 

1.949 

0.450 

2.923 

0.675 

3.897 

0.900 

4.872 

1.125 

77    0 

15 

0.973 

0.229 

1.947 

0.458 

2.920 

0.6SS 

3.894 

0.917 

4.867 

1.146 

45 

30 

0.972 

0.233 

1.945 

0.467 

2.917 

0.700 

3.SS9 

0.934 

4.S62 

1.167 

30 

45 

0.971 

0.238 

1.943 

0.475 

2.914 

0.713 

3.SS5 

0.951 

4.857 

1.188 

15 

14   0 

0.970 

0.242 

1.941 

0.484 

2.911 

0.726 

3.SS1 

0.968 

4.851 

1.210 

76    0 

15 

0.969 

0.246 

1.938 

0.492 

2.908 

0.73S 

3.877 

0.985 

4.846 

1.231 

45 

30 

0.968 

0.250 

1.936 

0.501 

2.904 

0.751 

3.873 

1.002 

4.841 

1.252 

30 

45 

0.967 

0.255 

1.934 

0.509 

2.901 

0.764 

3.868 

1.018 

4.835 

1.273 

15 

15   0 

0.966 

0.259 

1.932 

0.518 

2.898 

0.776 

3.864 

1.035 

4.830 

1.294 

75    0 

0      f 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

C        f 

Bearing 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

75°-  90' 


67 


Bearing, 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing. 

o    t 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

O          f 

015 

6.000 

0.026 

7.000 

0.031 

8.000 

0.035 

9.000 

0.039 

10.000 

0.044 

89  45 

30 

6.000 

0.052 

7.000 

0.061 

8.000 

0.070 

9.000 

0.079 

10.000 

0.087 

30 

45 

5.999 

0.079 

6.999 

0.092 

7.999 

0.105 

8.999 

0.118 

9.999 

0.131 

15 

1    0 

5.999 

0.105 

6.999 

0.122 

7.999 

0.140 

8.999 

0.157 

9.999 

0.175 

89    0 

15 

5.999 

0.131 

6.998 

0.153 

7.998 

0.175 

8.998 

0.196 

9.998 

0.218 

45 

30 

5.998 

0.157 

6.998 

0.183 

7.997 

0.209 

8.997 

0.236 

9.997 

0.262 

30 

45 

5.997 

0.183 

6.997 

0.214 

7.996 

0.244 

8.996 

0.275 

9.995 

0.305 

15 

2    0 

5.996 

0.209 

6.996 

0.244 

7.995 

0.279 

8.995 

0.314 

9.994 

0.349 

88    0 

15 

5.995 

0.236 

6.995 

0.275 

7.994 

0.314 

8.993 

0.353 

9.992 

0.393 

45 

30 

5.994 

0.262 

6.993 

0.305 

7.992 

0.349 

8.991 

0.393 

9.991 

0.436 

30 

45 

5.993 

0.288 

6.992 

0.336 

7.991 

0.384 

8.990 

0.432 

9.989 

0.480 

15 

3    0 

5.992 

0.314 

6.990 

0.366 

7.989 

0.419 

8.988 

0.471 

9.986 

0.523 

87    0 

15 

5.990 

0.340 

6.989 

0.397 

7.987 

0.454 

8.986 

0.510 

9.984 

0.567 

45 

30 

5.989 

0.366 

6.987 

0.427 

7.985 

0.488 

8.983 

0-549 

9.981 

0.611 

30 

45 

5.987 

0.392 

6.985 

0.458 

7.983 

0.523 

8.981 

0.589 

9.979 

0.654 

15 

4    0 

5.985 

0.419 

6.983 

0.488 

7.981 

0.558 

8.978 

0.628 

9.976 

0.698 

86   0 

15 

5.984 

0.445 

6.981 

0.519 

7.978 

0.593 

8.975 

0.667 

9.973 

0.741 

45 

30 

5.982 

0.471 

6.978 

0.549 

7.975 

0.628 

8.972 

0.706 

9.969 

0.785 

30 

45 

5.979 

0.497 

6.976 

0.580 

7.973 

0.662 

S.%9 

0.745 

9.966 

0.828 

15 

o    0 

5.977 

0.523 

6.973 

0.610 

7.970 

0.697 

8.966 

0.784 

9.962 

0.872 

85   0 

15 

5.975 

0.549 

6.971 

0.641 

7.966 

0.732 

8.962 

0.824 

9.958 

0.915 

45 

30 

5.972 

0.575 

6.968 

0.671 

7.963 

0.767 

8.959 

0.863 

9.954 

0.959 

30 

45 

5.970 

0.601 

6.965 

0.701 

7.960 

0.802 

8.955 

0.902 

9.950 

1.002 

15 

6    0 

5.967 

0.627 

6.962 

0.732 

7.956 

0.836 

8.951 

0.941 

9.945 

1.045 

84    0 

15 

5.964 

0.653 

6.958 

0.762 

7.952 

0.871 

8.947 

0.980 

9.941 

1.089 

45 

30 

5.961 

0.679 

6.955 

0.792 

7.949 

0.906 

8.942 

1.019 

9.936 

1.132 

30 

45 

5.958 

0.705 

6.951 

0.823 

7.945 

0.940 

8.938 

1.058 

9.931 

1.175 

15 

7    0 

5.955 

0.731 

6.948 

0.853 

7.940 

0.975 

8.933 

1.097 

9.926 

1.219 

83    0 

15 

5.952 

0.757 

6.944 

0.883 

7.936 

1.010 

8.928 

1.136 

9.920 

1.262 

45 

30 

5.949 

0.783 

6.940 

0.914 

7.932 

1.044 

8.923 

1.175 

9.914 

1.305 

30 

45 

5.945 

0.809 

6.936 

0.944 

7.927 

1.079 

8.918 

1.214 

9.909 

1.349 

15 

8    0 

5.942 

0.835 

6.932 

0.974 

7.922 

1.113 

8.912 

1.253 

9.903 

1.392 

82    0 

15 

5.938 

0.861 

6.928 

1.004 

7.917 

1.148 

8.907 

1.291 

9.897 

1.435 

45 

30 

5.934 

0.887 

6.923 

1.035 

7.912 

1.182 

8.901 

1.330 

9.890 

1.478 

30 

45 

5.930 

0.913 

6.919 

1.065 

7.907 

1.217 

8.895 

1.369 

9.884 

1.521 

15 

9    0 

5.926 

0.939 

6.914 

1.095 

7.902 

1.251 

8.889 

1.408 

9.877 

1.564 

81    0 

15 

5.922 

0.964 

6.909 

1.125 

7.896 

1.286 

8.883 

1.447 

9.870 

1.607 

45 

30 

5.918 

0.990 

6.904 

1.155 

7.890 

1.320 

8.877 

1.485 

9.863 

1.651 

30 

45 

5.913 

1.016 

6.899 

1.185 

7.884 

1.355 

8.870 

1.524 

9.856 

1.694 

16 

10    0 

5.909 

1.042 

6.894 

1.216 

7.878 

1.389 

8.863 

1.563 

9.848 

1.737 

80    0 

15 

5.904 

1.068 

6.888 

1.246 

7.872 

1.424 

8.856 

1.601 

9.840 

1.779 

45 

30 

5.900 

1.093 

6.883 

1.276 

7.866 

1.458 

8.849 

1.640 

9.833 

1.822 

30 

45 

5.895 

1.119 

6.877 

1.306 

7.860 

1.492 

8.842 

1.679 

9.825 

1.865 

15 

11    0 

5.890 

1.145 

6.871 

1.336 

7.853 

1.526 

8.835 

1.717 

9.816 

1.908 

79    0 

15 

5.885 

1.171 

6.866 

1.366 

7.846 

1.561 

8.827 

1.756 

9.808 

1.951 

45 

30 

5.880 

1.196 

6.859 

1.396 

7.839 

1.595 

8.S19 

1.794 

9.799 

1.994 

30 

45 

5.874 

1.222 

6.853 

1.425 

7.832 

1.629 

8.811 

1.833 

9.791 

2.036 

15 

12    0 

5.869 

1.247 

6.847 

1.455 

7.825 

1.663 

8.803 

1.871 

9.782 

2.079 

78    0 

15 

5.863 

1.273 

6.841 

1.485 

7.818 

1.697 

8.795 

1.910 

9.772 

2.122 

45 

30 

5.858 

1.299 

6.834 

1.515 

7.810 

1.732 

8.787 

1.948 

9.763 

2.164 

30 

45 

5.852 

1.324 

6.827 

1.545 

7.803 

1.766 

8.778 

1.986 

9.753 

2.207 

15 

13    0 

5.846 

1.350 

6.821 

1.575 

7.795 

1.800 

8.769 

2.025 

9.744 

2.250 

77    0 

15 

5.840 

1.375 

6.814 

1.604 

7.787 

1.834 

8.760 

2.063 

9.734 

2.292 

45 

30 

5.834 

1.401 

6.807 

1.634 

7.779 

1.868 

8.751 

2.101 

9.724 

2.335 

30 

45 

5.828 

1.426 

6.799 

1.664 

7.771 

1.902 

8.742 

2.139 

9.713 

2.377 

15 

14    0 

5.822 

1.452 

6.792 

1.693 

7.762 

1.935 

8.733 

2.177 

9.703 

2.419 

70    0 

15 

5.815 

1.477 

6.785 

1.723 

7.754 

1.969 

8.723 

2.215 

9.692 

2.462 

45 

30 

5.809 

1.502 

6.777 

1.753 

7.745 

2.003 

8.713 

2.253 

9.682 

2.504 

30 

45 

5.802 

1.528 

6.769 

1.782 

7.736 

2.037 

8.703 

2.291 

9.671 

2.546 

15 

15    0 

5.796 

1.553 

6.761 

1.812 

7.727 

2.071 

8.693 

2.329 

9.659 

2.588 

7.-,    0 

o    t 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o      t 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing. 

75°-  90 


58 


15°-  30C 


Bearing, 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Searing. 

o     t 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.    1   Dep. 

0         f 

1515 

0.965 

0.263 

1.930 

0.526 

2.894 

0.789 

3.859 

1.052 

4.824 

1.315 

7445 

30 

0.964 

0,267 

1.927 

0.534 

2.891 

0.802 

3.855 

1.069 

4.818 

1.336 

30 

45 

0.962 

0.271 

1.925 

0.543 

2.887 

0.814 

3.850 

1.086 

4.812 

1.357 

15 

16    0 

0.961 

0.276 

1.923 

0.551 

2.884 

0.827 

3.845 

1.103 

4.806 

1.378 

74    0 

15 

0.960 

0.280 

1.920 

0.560 

2.880 

O.S39 

3.840 

1.119 

4.800 

1.399 

45 

30 

0.959 

0.284 

1.918 

0.568 

2.876 

0.852 

3.835 

1.136 

4.794 

1.420 

30 

45 

0.958 

0.2SS 

1.915 

0.576 

2.873 

0.865 

3.830 

1.153 

4.788 

1.441 

15 

17    0 

0.956 

0.292 

1.913 

0.585 

2.869 

0.877 

3.825 

1.169 

4.7S2 

1.462 

73    0 

15 

0.955 

0.297 

1.910 

0.593 

2.865 

O.S90 

3.820 

1.186 

4.775 

1.483 

45 

30 

0.954 

0.301 

1.907 

0.601 

2.861 

0.902 

3.815 

1.203 

4.769 

1.504 

30 

45 

0.952 

0.305 

1.905 

0.610 

2.857 

0.915 

3.810 

1.220 

4.762 

1.524 

15 

18    0 

0.951 

0.309 

1.902 

0.618 

2.853 

0.927 

3.804 

1.236 

4.755 

1.545 

72    0 

15 

0.950 

0.313 

1.899 

0.626 

2.849 

0.939 

3.799 

1.253 

4.748 

1.566 

45 

30 

0.948 

0.317 

1.897 

0.635 

2.845 

0.952 

3.793 

1.269 

4.742 

1.587 

30 

45 

0.947 

0.321 

1.894 

0.643 

2.841 

0.964 

3.788 

1.286 

4.735 

r607 

15 

19    0 

0.946 

0.326 

1.891 

0.651 

2.837 

0.977 

3.782 

1.302 

4.728 

1.628 

71    0 

15 

0.944 

0.330 

1.888 

0.659 

2.832 

0.989 

3.776 

1.319 

4.720 

1.648 

45 

30 

0.943 

0.334 

1.885 

0.668 

2.828 

1.001 

3.771 

1.335 

4.713 

1.669 

30 

45 

0.941 

0.338 

1.882 

0.676 

2.824 

1.014 

3.765 

1.352 

4.706 

1.690 

15 

2O  0 

0.940 

0.342 

1.879 

0.684 

2.819 

1.026 

3.759 

1.368 

4.698 

1.710 

70    0 

15 

0.938 

0.346 

1.876 

0.692 

2.815 

1.038 

3.753 

1.384 

4.691 

1.731 

45 

30 

0.937 

0.350 

1.873 

0.700 

2.810 

1.051 

3.747 

1.401 

4.683 

1.751 

30 

45 

0.935 

0.354 

1.870 

0.709 

2.805 

1.063 

3.741 

1.417 

4.676 

1.771 

15 

21    0 

0.934 

0.358 

1.867 

0.717 

2.801 

1.075 

3.734 

1.433 

4.668 

1.792 

69    0 

15 

0.932 

0.362 

1.864 

0.725 

2.796 

1.087 

3.728 

1.450 

4.660 

1.812 

45 

30 

0.930 

0.367 

1.861 

0.733 

2.791 

1.100 

3.722 

1.466 

4.652 

1.833 

30 

45 

0.929 

0.371 

1.858 

0.741 

2.786 

1.112 

3.715 

1.482 

4.644 

1.853 

15 

22  0 

0.927 

0.375 

1.854 

0.749 

2.782 

1.124 

3.709 

1.498 

4.636 

1.873 

68    0 

15 

0.926 

0.379 

1.851 

0.757 

2.777 

1.136 

3.702 

1.515 

4.628 

1.893 

45 

30 

0.924 

0.383 

1.848 

0.765 

2.772 

1.148 

3.696 

1.531 

4.619 

1.913 

30 

45 

0.922 

0.387 

1.844 

0.773 

2.767 

1.160 

3.689 

1.547 

4.611 

1.934 

15 

23  0 

0.921 

0.391 

1.841 

0.781 

2.762 

1.172 

3.682 

1.563 

4.603 

1.954 

67    0 

15 

0.919 

0.395 

1.838 

0.789 

2.756 

1.184 

3.675 

1.579 

4.594 

1.974 

45 

30 

0.917 

0.399 

1.834 

0.797 

2.751 

1.196 

3.668 

1.595 

4.585 

1.994 

30 

45 

0.915 

0.403 

1.831 

0.805 

2.746 

1.208 

3.661 

1.611 

4.577 

2.014 

15 

24  0 

0.914 

0.407 

1.827 

0.813 

2.741 

1.220 

3.654 

1.627 

4.568 

2.034 

66    0 

15 

0.912 

0.411 

1.824 

0.821 

2.735 

1.232 

3.647 

1.643 

4.559 

2.054 

45 

30 

0.910 

0.415 

1.820 

0.829 

2.730 

1.244 

3.640 

1.659 

4.550 

2.073 

30 

45 

0.908 

0.419 

1.816 

0.837 

2.724 

1.256 

3.633 

1.675 

4.541 

2.093 

15 

25  0 

0.906 

0.423 

1.813 

0.845 

2.719 

1.268 

3.625 

1.690 

4.532 

2.113 

65   0 

15 

0.904 

0.427 

1.809 

0.853 

2.713 

1.280 

3.618 

1.706 

4.522 

2.133 

45 

30 

0.903 

0.431 

1.805 

0.861 

2.708 

1.292 

3.610 

1.722 

4.513 

2.153 

30 

45 

0.901 

0.434 

1.801 

0.869 

2.702 

1.303 

3.603 

1.738 

4.503 

2.172 

15 

26  0 

0.899 

0.438 

1.798 

0.877 

2.696 

1.315 

3.595 

1.753 

4.494 

2.192 

64    0 

15 

0.897 

0.442 

1.794 

0.885 

2.691 

1.327 

3.587 

1.769 

4.484 

2.211 

45 

30 

0.895 

0.446 

1.790 

0.892 

2.685 

1.339 

3.580 

1.785 

4.475 

2.231 

30 

45 

0.893 

0.450 

1.786 

0.900 

2.679 

1.350 

3.572 

1.800 

4.465 

2.250 

15 

27   0 

0.891 

0.454 

1.782 

0.908 

2.673 

1.362 

3.564 

1.816 

4.455 

2.270 

63    0 

15 

0.889 

0.458 

1.778 

0.916 

2.667 

1.374 

3.556 

1.831 

4.445 

2.289 

45 

30 

0.887 

0.462 

1.774 

0.923 

2.661 

1.385 

3.548 

1.847 

4.435 

2.309 

30 

45 

0.885 

0.466 

1.770 

0.931 

2.655 

1.397 

3.540 

1.862 

4.425 

2.328 

15 

28  0 

0.883 

0.469 

1.766 

0.939 

2.649 

1.408 

3.532 

1.878 

4.415 

2.347 

62    0 

15 

0.881 

0.473 

1.762 

0.947 

2.643 

1.420 

3.524 

1.893 

4.404 

2.367 

45 

30 

0.879 

0.477 

1.758 

0.954 

2.636 

1.431 

3.515 

1.909 

4.394 

2.386 

30 

45 

0.877 

0.481 

1.753 

0.962 

2.630 

1.443 

3.507 

1.924 

4.384 

2.405 

15 

29  0 

0.875 

0.485 

1.749 

0.970 

2.624 

1.454 

3.498 

1.939 

4.373 

2.424 

61    0 

15 

0.872 

0.489 

1.745 

0.977 

2.617 

1.466 

3.490 

1.954 

4.362 

2.443 

45 

30 

0.870 

0.492 

1.741 

0.985 

2.611 

1.477 

3.481 

1.970 

4.352 

2.462 

30 

45 

0.868 

0.496 

1.736 

0.992 

2.605 

1.489 

3.473 

1.985 

4.341 

2.481 

15 

30   0 

0.866 

0.500 

1.732 

1.000 

2.598 

1.500 

3.464 

2.000 

4.330 

2.500 

6O    0 

0      f 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

0        t 

Bearing 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

60°-  75< 


15°-  30' 


59 


Bearing, 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  10. 

Bearing. 

O       f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o     t 

1515 

5.789 

1.578 

6.754 

1.841 

7.718 

2.104 

8.683 

2.367 

9.648 

2.630 

7445 

30 

5.782 

1.603 

6.745 

1.871 

7.709 

2.138 

8.673 

2.405 

9.636 

2.672 

30 

45 

5.775 

1.629 

6.737 

1.900 

7.700 

2.172 

8.662 

2.443 

9.625 

2.714 

15 

16    0 

5.768 

1.654 

6.729 

1.929 

7.690 

2.205 

8.651 

2.481 

9.613 

2.756 

74   0 

15 

5.760 

1.679 

6.720 

1.959 

7.680 

2.239 

8.640 

2.518 

9.601 

2.798 

45 

30 

5.753 

1.704 

6.712 

1.988 

7.671 

2.272 

8.629 

2.556 

9.588 

2.840 

30 

45 

5.745 

1.729 

6.703 

2.017 

7.661 

2.306 

8.618 

2.594 

9.576 

2.882 

15 

17    0 

5.738 

1.754 

6.694 

2.047 

7.650 

2.339 

8.607 

2.631 

9.563 

2.924 

73    0 

15 

5.730 

1.779 

6.685 

2.076 

7.640 

2.372 

8.595 

2.669 

9.550 

2.965 

45 

30 

5.722 

1.804 

6.676 

2.105 

7.630 

2.406 

8.583 

2.706 

9.537 

3.007 

30 

45 

5.714 

1.829 

6.667 

2.134 

7.619 

2.439 

8.572 

2.744 

9.524 

3.049 

15 

18    0 

5.706 

1.854 

6.657 

2.163 

7.608 

2.472 

8.560 

2.781 

9.511 

3.090 

72   0 

15 

5.698 

1.879 

6.648 

2.192 

7.598 

2.505 

8.547 

2.818 

9.497 

3.132 

45 

30 

5.690 

1.904 

6.638 

2.221 

7.587 

2.538 

8.535 

2.856 

9.483 

3.173 

30 

45 

5.682 

1.929 

6.629 

2.250 

7.575 

2.572 

8.522 

2.893 

9.469 

3.214 

15 

19    0 

5.673 

1.953 

6.619 

2.279 

7.564 

2.605 

8.510 

2.930 

9.455 

3.256 

71    0 

15 

5.665 

1.978 

6.609 

2.308 

7.553 

2.638 

8.497 

2.967 

9.441 

3.297 

45 

30 

5.656 

2.003 

6.598 

2.337 

7.541 

2.670 

8.484 

3.004 

9.426 

3.338 

30 

45 

5.647 

2.028 

6.588 

2.365 

7.529 

2.703 

8.471 

3.041 

9.412 

3.379 

15 

20  0 

5.638 

2.052 

6.578 

2.394 

7.518 

2.736 

8.457 

3.078 

9.397 

3.420 

70   0 

15 

5.629 

2.077 

6.567 

2.423 

7.506 

2.769 

8.444 

3.115 

9.382 

3.461 

45 

30 

5.620 

2.101 

6.557 

2.451 

7.493 

2.802 

8.430 

3.152 

9.367 

3.502 

30 

45 

5.611 

2.126 

6.546 

2.480 

7.481 

2.834 

8.416 

3.189 

9.351 

3.543 

15 

21    0 

5.601 

2.150 

6.535 

2.509 

7.469 

2.867 

8.402 

3.225 

9.336 

3.584 

69    0 

15 

5.592 

2.175 

6.524 

2.537 

7.456 

2.900 

8.388 

3.262 

9.320 

3.624 

45 

30 

5.582 

2.199 

6.513 

2.566 

7.443 

2.932 

8.374 

3.299 

9.304 

3.665 

30 

45 

5.573 

2.223 

6.502 

2.594 

7.430 

2.964 

8.359 

3.335 

9.288 

3.706 

15 

22   0 

5.563 

2.248 

6.490 

2.622 

7.417 

2.997 

8.345 

3.371 

9.272 

3.746 

68    0 

15 

5.553 

2.272 

6.479 

2.651 

7.404 

3.029 

8.330 

3.408 

9.255 

3.787 

45 

30 

5.543 

2.296 

6.467 

2.679 

7.391 

3.061 

8.315 

3.444 

9.239 

3.827 

30 

45 

5.533 

2.320 

6.455 

2.707 

7.378 

3.094 

8.300 

3.480 

9.222 

3.867 

15 

23   0 

5.523 

2.344 

6.444 

2.735 

7.364 

3.126 

8.285 

3.517 

9.205 

3.907 

67    0 

15 

5.513 

2.368 

6.432 

2.763 

7.350 

3.158 

8.269 

3.553 

9.188 

3.947 

45 

30 

5.502 

2.392 

6.419 

2.791 

7.336 

3.190 

8.254 

3.589 

9.171 

3.988 

30 

45 

5.492 

2.416 

6.407 

2.819 

7.322 

3.222 

8.238 

3.625 

9.153 

4.028 

15 

24  0 

5.481 

2.440 

6.395 

2.847 

7.308 

3.254 

8.222 

3.661 

9.136 

4.067 

66    0 

15 

5.471 

2.464 

6.382 

2.875 

7.294 

3.286 

8.206 

3.696 

9.118 

4.107 

45 

30 

5.460 

2.488 

6.370 

2.903 

7.280 

3.318 

8.190 

3.732 

9.100 

4.147 

30 

45 

5.449 

2.512 

6.357 

2.931 

7.265 

3.349 

8.173 

3.768 

9.081 

4.187 

15 

25  0 

5.438 

2.536 

6.344 

2.958 

7.250 

3.381 

8.157 

3.804 

9.063 

4.226 

65    0 

15 

5.427 

2.559 

6.331 

2.986 

7.236 

3.413 

8.140 

3.839 

9.045 

4.266 

45 

30 

5.416 

2.583 

6.318 

3.014 

7.221 

3.444 

8.123 

3.875 

9.026 

4.305 

30 

45 

5.404 

2.607 

6.305 

3.041 

7.206 

3.476 

8.106 

3.910 

9.007 

4.345 

15 

26  0 

5.393 

2.630 

6.292 

3.069 

7.190 

3.507 

8.089 

3.945 

8.988 

4.384 

64   0 

15 

5.3S1 

2.654 

6.278 

3.096 

7.175 

3.538 

8.072 

3.981 

8.969 

4.423 

45 

30 

5.370 

2.677 

6.265 

3.123 

7.160 

3.570 

8.054 

4.016 

8.949 

4.462 

30 

45 

5.358 

2.701 

6.251 

3.151 

7.144 

3.601 

8.037 

4.051 

8.930 

4.501 

15 

27   0 

5.346 

2.724 

6.237 

3.178 

7.128 

3.632 

8.019 

4.086 

8.910 

4.540 

63    0 

15 

5.334 

2.747 

6.223 

3.205 

7.112 

3.663 

8.001 

4.121 

8.890 

4.579 

45 

30 

5.322 

2.770 

6.209 

3.232 

7.096 

3.694 

7.983 

4.156 

8.870 

4.618 

30 

45 

5.310 

2.794 

6.195 

3.259 

7.0SO 

3.725 

7.965 

4.190 

8.850 

4.656 

15 

28  0 

5.298 

2.817 

6.181 

3.286 

7.064 

3.756 

7.947 

4.225 

8.829 

4.695 

62    0 

15 

5.285 

2.840 

6.166 

3.313 

7.047 

3.787 

7.928 

4.260 

8.809 

4.733 

45 

30 

5.273 

2.863 

6.152 

3.340 

7.031 

3.817 

7.909 

4.294 

8.788 

4.772 

30 

45 

5.260 

2.886 

6.137 

3.367 

7.014 

3.848 

7.891 

4.329 

8.767 

4.810 

15 

29  0 

5.248 

2.909 

6.122 

3.394 

6.997 

3.878 

7.S72 

4.363 

8.746 

4.848 

61    0 

15 

5.235 

2.932 

6.107 

3.420 

6.9SO 

3.909 

7.852 

4.398 

8.725 

4.886 

45 

30 

5.222 

2.955 

6.093 

3.447 

6.963 

3.939 

7.833 

4.432 

8.704 

4.924 

30 

45 

5.209 

2.977 

6.077 

3.474 

6.946 

3.970 

7.814 

4.466 

8.682 

4.962 

15 

30   0 

5.196 

3.000 

6.062 

3.500 

6.928 

4.000 

7.794 

4.500 

8.660 

5.000 

60    0 

0      f 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o     t 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing. 

60°--  75' 


30°-  45C 


Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

o    r 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

0         f 

3015 

0.864 

0.504 

1.728 

1.008 

2.592 

1.511 

3.455 

2.015 

4.319 

2.519 

5945 

30 

0.862 

0.508 

1.723 

1.015 

2.585 

1.523 

3.447 

2.030 

4.308 

2.538 

30 

45 

0.859 

0.511 

1.719 

1.023 

2.578 

1.534 

3.438 

2.045 

4.297 

2.556 

15 

31    0 

0.857 

0.515 

1.714 

1.030 

2.572 

1.545 

3.429 

2.060 

4.286 

2.575 

59    0 

15 

0.855 

0.519 

1.710 

1.038 

2.565 

1.556 

3.420 

2.075 

4.275 

2.594 

45 

30 

0.853 

0.522 

1.705 

1.045 

2.558 

1.567 

3.411 

2.090 

4.263 

2.612 

30 

45 

0.850 

0.526 

1.701 

1.052 

2.551 

1.579 

3.401 

2.105 

4.252 

2.631 

15 

32  0 

0.848 

0.530 

1.696 

1.060 

2.544 

1.590 

3.392 

2.120 

4.240 

2.650 

58    0 

15 

0.846 

0.534 

1.691 

1.067 

2.537 

1.601 

3.383 

2.134 

4.229 

2.668 

45 

30 

0.843 

0.537 

1.687 

1.075 

2.530 

1.612 

3.374 

2.149 

4.217 

2.686 

30 

45 

0.841 

0.541 

1.682 

1.082 

2.523 

1.623 

3.364 

2.164 

4.205 

2.705 

15 

33  0 

0.839 

0.545 

1.677 

1.089 

2.516 

1.634 

3.355 

2.179 

4.193 

2.723 

57    0 

15 

0.836 

0.548 

1.673 

1.097 

2.509 

1.645 

3.345 

2.193 

4.181 

2.741 

45 

30 

0.834 

0.552 

1.668 

1.104 

2.502 

1.656 

3.336 

2.208 

4.169 

2.760 

30 

45 

0.831 

0.556 

1.663 

1.111 

2.494 

1.667 

3.326 

2.222 

4.157 

2.778 

15 

34  0 

0.829 

0.559 

1.658 

1.118 

2.487 

1.678 

3.316 

2.237 

4.145 

2.796 

56    0 

15 

0.827 

0.563 

1.653 

1.126 

2.480 

1.688 

3.306 

2.251 

4.133 

2.814 

45 

30 

0.824 

0.566 

1.648 

1.133 

2.472 

1.699 

3.297 

2.266 

4.121 

2.832 

30 

45 

0.822 

0.570 

1.643 

1.140 

2.465 

1.710 

3.287 

2.280 

4.108 

2.850 

15 

35  0 

0.819 

0.574 

1.638 

1.147 

2.457 

1.721 

3.277 

2.294 

4.096 

2.868 

55   0 

15 

0.817 

0.577 

1.633 

1.154 

2.450 

1.731 

3.267 

2.309 

4.083 

2.886 

45 

30 

0.814 

0.581 

1.628 

1.161 

2.442 

1.742 

3.257 

2.323 

4.071 

2.904 

30 

45 

0.812 

0.584 

1.623 

1.168 

2.435 

1.753 

3.246 

2.337 

4.058 

2.921 

15 

36  0 

0.809 

0.588 

1.618 

1.176 

2.427 

1.763 

3.236 

2.351 

4.045 

2.939 

54    0 

15 

0.806 

0.591 

1.613 

1.183 

2.419 

1.774 

3.226 

2.365 

4.032 

2.957 

45 

30 

0.804 

0.595 

1.608 

1.190 

2.412 

1.784 

3.215 

2.379 

4.019 

2.974 

30 

45 

0.801 

0.598 

1.603 

1.197 

2.404 

1.795 

3.205 

2.393 

4.006 

2.992 

15 

37  0 

0.799 

0.602 

1.597 

1.204 

2.396 

1.805 

3.195 

?.407 

3.993 

3.009 

53    0 

15 

0.796 

0.605 

1.592 

1.211 

2.388 

1.816 

3.184 

2.421 

3.980 

3.026 

45 

30 

0.793 

0.609 

1.587 

1.218 

2.380 

1.826 

3.173 

2.435 

3.967 

3.044 

30 

45 

0.791 

0.612 

1.581 

1.224 

2.372 

1.837 

3.163 

2.449 

3.953 

3.061 

15 

38  0 

0.788 

0.616 

1.576 

1.231 

2.364 

1.847 

3.152 

2.463 

3.940 

3.078 

52    0 

15 

0.785 

0.619 

1.571 

1.238 

2.356 

1.857 

3.141 

2.476 

3.927 

3.095 

45 

30 

0.783 

0.623 

1.565 

1.245 

2.348 

1.868 

3.130 

2.490 

3.913 

3.113 

30 

45 

0.780 

0.626 

1.560 

1.252 

2.340 

1.878 

3.120 

2.504 

3.899 

3.130 

15 

39  0 

0.777 

0.629 

1.554 

1.259 

2.331 

1.888 

3.109 

2.517 

3.8S6 

3.147 

51    0 

15 

0.774 

0.633 

1.549 

1.265 

2.323 

1.898 

3.098 

2.531 

3.872 

3.164 

45 

30 

0.772 

0.636 

1.543 

1.272 

2.315 

1.908 

3.086- 

2.544 

3.858 

3.180 

30 

45 

0.769 

0.639 

1.538 

1.279 

2.307 

1.918 

3.075 

2.558 

3.  844 

3.197 

15 

40  0 

0.766 

0.643 

1.532 

1.286 

2.298 

1.928 

3.064 

2.571 

3.830 

3.214 

5O    0 

15 

0.763 

0.646 

1.526 

1.292 

2.290 

1.938 

3.053 

2.584 

3.816 

3.231 

45 

30 

0.760 

0.649 

1.521 

1.299 

2.281 

1.948 

3.042 

2.598 

3.802 

3.247 

30 

45 

0.758 

0.653 

1.515 

1.306 

2.273 

1.958 

3.030 

2.611 

3.  788 

3.264 

15 

41    0 

0.755 

0.656 

1.509 

1.312 

2.264 

1.968 

3.019 

2.624 

3.774 

3.280 

49    0 

15 

0.752 

0.659 

1.504 

1.319 

2.256 

1.978 

3.007 

2.637 

3.759 

3.297 

45 

30 

0.749 

0.663 

1.498 

1.325 

2.247 

1.988 

2.996 

2.650 

3.745 

3.313 

30 

45 

0.746 

0.666 

1.492 

1.332 

2.23S 

1.998 

2.984 

2.664 

3.730 

3.329 

15 

42  0 

0.743 

0.669 

1.486 

1.338 

2.229 

2.007 

2.973 

2.677 

3.716 

3.346 

48    0 

15 

0.740 

0.672 

1.480 

1.345 

2.221 

2.017 

2.961 

2.689 

3.701 

3.362 

45 

30 

0.737 

0.676 

1.475 

1.351 

2.212 

2.027 

2.949 

2.702 

3.686 

3.378 

30 

45 

0.734 

0.679 

1.469 

1.358 

2.203 

2.036 

2.937 

2.715 

3.672 

3.394 

15 

43  0 

0.731 

0.682 

1.463 

1.364 

2.194 

2.046 

2.925 

2.728 

3.657 

3.410 

47    0 

15 

0.728 

0.685 

1.457 

1.370 

2.185 

2.056 

2.913 

2.741 

3.642 

3.426 

45 

30 

0.725 

0.688 

1.451 

1.377 

2.176 

2.065 

2.901 

2.753 

3.627 

3.442 

30 

45 

0.722 

0.692 

1.445 

1.383 

2.167 

2.075 

2.889 

2.766 

3.612 

3.458 

15 

44  0 

0.719 

0.695 

1.439 

1.389 

2.158 

2.0S4 

2.877 

2.779 

3.597 

3.473 

46    0 

15 

0.716 

0.698 

1.433 

1.396 

2.149 

2.093 

2.865 

2.791 

3.582 

3.489 

45 

30 

0.713 

0.701 

1.427 

1.402 

2.140 

2.103 

2.853 

2.804 

3.566 

3.505 

30 

45 

0.710 

0.704 

1.420 

1.408 

2.131 

2.112 

2.841 

2.816 

3.551 

3.520 

15 

45  0 

0.707 

0.707 

1.414 

1.414 

2.121 

2.121 

2.828 

2.828 

3.536 

3.536 

45   0 

0      f 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o    r 

Bearing 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

45°-  60C 


30°-  45' 


Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing. 

o     / 

Lat.       Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.    i   Dep. 

o      t 

3O15 

5.183 

3.023 

6.047 

3.526 

6.911 

4.030 

7.775 

4.534 

8.638  i  5.038 

5945 

30 

5.170 

3.045 

6.031 

3.553 

6.893 

4.060 

7.755 

4.568 

8.616    5.075 

30 

45 

5.156 

3.  068 

6.016 

3.579 

6.875 

4.090 

7.735 

4.602 

8.594 

5.113 

15 

31    0 

5.143 

3.090 

6.000 

3.605 

6.857 

4.120 

7.715 

4.635 

8.572 

5.150 

59    0 

15 

5.129 

3.113 

5.984 

3.631 

6.839 

4.150 

7.694 

4.669 

8.549 

5.188 

45 

30 

5.116 

3.135 

5.968 

3.657 

6.821 

4.180 

7.674 

4.702 

8.526 

5.225 

30 

45 

5.102 

3.157 

5.952 

3.683 

6.803 

4.210 

7.653 

4.736 

8.504 

5.262 

15 

32   0 

5.088 

3.180 

5.936 

3.709 

6.784 

4.239 

7.632 

4.769 

8.481 

5.299 

58   0 

15 

5.074 

3.202 

5.920 

3.735 

6.766 

4.269 

7.612 

4.802 

8.457 

5.336 

45 

30 

5.060 

3.224 

5.904 

3.761 

6.747 

4.298 

7.591 

4.836 

8.434 

5.373 

30 

45 

5.046 

3.246 

5.887 

3.787 

6.728 

4.328 

7.569 

4.869 

8.410 

5.410 

15 

33   0 

5.032 

3.268 

5.871 

3.812 

6.709 

4.357 

7.548 

4.902 

8.387 

5.446 

57    0 

15 

5.018 

3.290 

5.854 

3.838 

6.690 

4.386 

7.527 

4.935 

8.363 

5.  483 

45 

30 

5.003 

3.312 

5.837 

3.864 

6.671 

4.416 

7.505 

4.967 

8.339 

5.519 

30 

45 

4.989 

3.333 

5.820 

3.889 

6.652 

4.445 

7.483 

5.000 

8.315 

5.556 

15 

34   0 

4.974 

3.355 

5.803 

3.914  j!  6.632 

4.474 

7.461 

5.033 

8.290 

5.592 

56    0 

15 

4.960 

3.377 

5.786 

3.940 

6.613 

4.502 

7.439 

5.065 

8.266 

5.628 

45 

30 

4.945 

3.398 

5.769 

3.965 

6.593 

4.531 

7.417 

5.098 

8.241 

5.664 

30 

45 

4.930 

3.420 

5.752 

3.990 

6.573 

4.560 

7.395 

5.130 

8.217 

5.700 

15 

35  0 

4.915 

3.441 

5.734 

4.015 

6.553 

4.589 

7.372 

5.162 

8.192 

5.736 

55   0 

15 

4.900 

3.463 

5.716 

4.040 

6.533 

4.617 

7.350 

5.194 

8.166 

5.772 

45 

30 

4.885 

3.484 

5.699 

4.065 

6.513 

4.646 

7.327 

5.226 

8.141 

5.807 

30 

45 

4.869 

3.505 

5.681 

4.090 

6.493 

4.674 

7.304 

5.258 

8.116 

5.843 

15 

36   0 

4.854 

3.527 

5.663 

4.115 

6-472 

4.702 

7.281 

5.290 

8.090 

5.878 

54   0 

15 

4.839 

3.548 

5.645 

4.139 

6.452 

4.730 

7.258 

5.322 

8.064 

5.913 

45 

30 

4.823 

3.569 

5.627 

4.164 

6.431 

4.759 

7.235 

5.353 

8.039 

5.948 

30 

45 

4.808 

3.590 

5.609 

4.188 

6.410 

4.787 

7.211 

5.385 

8.013 

5.983 

15 

37  0 

4.792 

3.611 

5.590 

4.213 

6.389 

4.815 

7.188 

5.416 

7.986 

6.018 

53    0 

15 

4.776 

3.632 

5.572 

4.237 

6.368 

4.842 

7.164 

5.448 

7.960 

6.053 

45 

30 

4.760 

3.653 

5.554 

4.261 

6.347 

4.870 

7.140 

5.479 

7.934 

6.088 

30 

45 

4.744 

3.673 

5.535 

4.286 

6.326 

4.898 

7.116 

5.510 

7.907 

6.122 

15 

38  0 

4.728 

3.694 

5.516 

4.310 

6.304 

4.925 

7.092 

5.541 

7.880 

6.157 

52    0 

15 

4.712 

3.715 

5.497 

4.334 

6.283 

4.953 

7.068 

5.572 

7.853 

6.191 

45 

30 

4.696 

3.735 

5.478 

4.358 

6.261 

4.980 

7.043 

5.603 

7.826 

6.225 

30 

45 

4.679 

3.756 

5.459 

4.381 

6.239 

5.007 

7.019 

5.633 

7.799 

6.259 

15 

39   0 

4.663 

3.776 

5.440 

4.405 

6.217 

5.035 

6.994 

5.664 

7.772 

6.293 

51    0 

15 

4.646 

3.796 

5.421 

4.429 

6.195 

5.062 

6.970 

5.694 

7.744 

6.327 

45 

30 

4.630 

3.816 

5.401 

4.453 

6.173 

5.089 

6.945 

5.725 

7.716 

6.361 

30 

45 

4.613 

3.837 

5.382 

4.476 

6.151 

5.116 

6.920 

5.755 

7.688 

6.394 

15 

40   0 

4.596 

3.857 

5.362 

4.500 

6.128 

5.142 

6.894 

5.785 

7.660 

6.428 

50    0 

15 

4.579 

3.877 

5.343 

4.523 

6.106 

5.169 

6.869 

5.815 

7.632 

6.461 

45 

30 

4.562 

3.897 

5.323 

4.546 

6.083 

5.196 

6.844 

5.845 

7.604 

6.495 

30 

45 

4.545 

3.917 

5.303 

4.569 

6.061 

5.222 

6.818 

5.875 

7.576 

6.528 

15 

41    0 

4.528 

3.936 

5.283 

4.592 

6.038 

5.248 

6.792 

5.905 

7.547 

6.561 

49    0 

15 

4.511 

3.956 

5.263 

4.615 

6.015 

5.275 

6.767 

5.934 

7.518 

6.594 

45 

30 

4.494 

3.976 

5.243 

4.638 

5.992 

5.301 

6.741 

5.964 

7.490 

6.626 

30 

45 

4.476 

3.995 

5.222 

4.661 

5.968 

5.327 

6.715 

5.993 

7.461 

6.659 

15 

42   0 

4.459 

4.015 

5.202 

4.684 

5.945 

5.353 

6.688 

6.022 

7.431 

6.691 

48    0 

15 

4.441 

4.034 

5.182 

4.707 

5.922 

5.379 

6.662 

6.051 

7.402 

6.724 

45 

30 

4.424 

4.054 

5.161 

4.729 

5.898 

5.405 

6.635 

6.080 

7.373 

6.756 

30 

45 

4.406 

4.073 

5.140 

4.752 

5.875 

5.430 

6.609 

6.109 

7.343 

6.788 

15 

43  0 

4.388 

4.092 

5.119 

4.774 

5.851 

5.456 

6.582 

6.138 

7.314 

6.820 

47    0 

15 

4.370 

4.111 

5.099 

4.7% 

5.827 

5.481 

6.555 

6.167 

7.284 

6.852 

45 

30 

4.352 

4.130 

5.078 

4.818 

5.803 

5.507 

6.528 

6.195 

7.254 

6.884 

30 

45 

4.334 

4.149 

5.057 

4.841 

5.779 

5.532 

6.501 

6.224 

7.224 

6.915 

15 

44  0 

4.316 

4.168 

5.035 

4.863 

5.755 

5.557 

6.474 

6.252 

7.193 

6.947 

48    0 

15 

4.298 

4.187 

5.014 

4.885 

5.730 

5.582 

6.447 

6.280 

7.163 

6.978 

45 

30 

4.2SO 

4.206 

4.993 

4.906 

5.706 

5.607 

6.419 

6.308 

7.133 

7.009 

30 

45 

4.261 

4.224 

4.971 

4.928 

5.681 

5.632 

6.392 

6.336 

7.102 

7.040 

15 

45  0 

4.243 

4.243 

4.950 

4.950 

5.657 

5.657 

6.364 

6.364 

7.071 

7.071 

45    0 

0      f 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o     t 

Bearing, 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing. 

45°-  60' 


TABLE  VIII, -NATURAL  SINES  AND  COSINES. 


t 

0° 

1° 

2° 

3° 

4° 

t 

sin   cos 

gin   cos 

sin   cos 

sin   cos 

sin   cos 

o 

0000   one 

0175  9998 

0349  9994 

0523  9986 

0698  9976 

6O 

1 

0003   one 

0177  9998 

0352  9994 

0526  9986 

0700  9975 

59 

2 

0006   one 

0180  9998 

0355  9994 

0529  9986 

0703  9975 

58 

3 

0009   one 

0183  9998 

0358  9994 

0532  9986 

0706  9975 

57 

4 

0012   one 

0186  9998 

0361  9993 

0535  9986 

0709  9975 

56 

5 

0015   one 

0189  9998 

0364  9993 

0538  9986 

0712  9975 

55 

6 

0017   one 

0192  9998 

0366  9993 

0541  9985 

0715  9974 

54 

7 

0020   one 

0195  9998 

0369  9993 

0544  9985 

0718  9974 

53 

8 

0023   one 

0198  9998 

0372  9993 

0547  9985 

0721  9974 

52 

9 

0026   one 

0201  9998 

0375  9993 

0550  9985 

0724  9974 

51 

1O 

0029   one 

0204  9998 

0378  9993 

0552  9985 

0727  9974 

50 

11 

0032   one 

0207  9998 

0381  9993 

0555  9985 

0729  9973 

49 

12 

0035   one 

0209  9998 

0384  9993 

0558  9984 

0732  9973 

48 

13 

0038   one 

0212  9998 

0387  9993 

0561  9984 

0735  9973 

47 

14 

0041   one 

0215  9998 

0390  9992 

0564  9984 

0738  9973 

46 

15 

0044  one 

0218  9998 

0393  9992 

0567  9984 

0741  9973 

45 

16 

0047  one 

0221  9998 

0396  9992 

0570  9984 

0744  9972 

44 

17 

0049   one 

0224  9997 

0398  9992 

0573  9984 

0747  9972 

43 

18 

0052   one 

0227  9997 

0401  9992 

0576  9983 

0750  9972 

42 

19 

0055   one 

0230  9997 

0404  9992 

0579  9983 

0753  9972 

41 

2O 

0058  one 

0233  9997 

0407  9992 

0581  9983 

0756  9971 

40 

21 

0061   one 

0236  9997 

0410  9992 

0584  9983 

0758  9971 

39 

22 

0064  one 

0239  9997 

0413  9991 

0587  9983 

0761  9971 

38 

23 

0067  one 

0241  9997 

0416  9991 

0590  9983 

0764  9971 

37 

24 

0070  one 

0244  9997 

0419  9991 

0593  9982 

0767  9971 

36 

25 

0073  one 

0247  9997 

0422  9991 

05%  9982 

0770  9970 

35 

26 

0076  one 

0250  9997 

0425  9991 

0599  9982 

0773  9970 

34 

27 

0079  one 

0253  9997 

0427  9991 

0602  9982 

0776  9970 

33 

28 

0081   one 

0256  9997 

0430  9991 

0605  9982 

0779  9970 

32 

29 

0084  one 

0259  9997 

0433  9991 

0608  9982 

0782  9969 

31 

30 

0087  one 

0262  9997 

0436  9990 

0610  9981 

0785  9969 

3O 

31 

0090  one 

0265  9996 

0439  9990 

0613  9981 

0787  9969 

29 

32 

0093  one 

0268  9996 

0442  9990 

0616  9981 

0790  9969 

28 

33 

0096   one 

0270  9996 

0445  9990 

0619  9981 

0793  9968 

27 

34 

0099  one 

0273  9996 

0448  9990 

0622  9981 

0796  9968 

26 

35 

0102  9999 

0276  9996 

0451  9990 

0625  9980 

0799  9968 

25 

36 

0105  9999 

0279  9996 

0454  9990 

0628  9980 

OS02  9968 

24 

37 

0108  9999 

0282  9996 

0457  9990 

0631  9980 

0805  9968 

23 

38 

0111  9999 

0285  9996 

0459  9989 

0634  9980 

0808  9967 

22 

39 

0113  9999 

0288  9996 

0462  9989 

0637  9980 

0811  9967 

21 

4O 

0116  9999 

0291  9996 

0465  9989 

0640  9980 

0814  9967 

2O 

41 

0119  9999 

0294  9996 

0468  9989 

0642  9979 

0816  9967 

19 

42 

0122  9999 

0297  9996 

0471  9989 

0645  9979 

0819  9966 

18 

43 

0125  9999 

0300  9996 

0474  9989 

0648  9979 

0822  9966 

17 

44 

0128  9999 

0302  9995 

0477  9989 

0651  9979 

0825  9966 

16 

45 

0131  9999 

0305  9995 

0480  9988 

0654  9979 

0828  9966 

15 

46 

0134  9999 

0308  9995 

0483  9988 

0657  9978 

0831  9965 

14 

47 

0137  9999 

0311  9995 

0486  9988 

0660  9978 

0834  9965 

13 

48 

0140  9999 

0314  9995 

0488  9988 

0663  9978 

0837  9965 

12 

49 

0143  9999 

0317  9995 

0491  9988 

0666  9978 

08-10  9965 

11 

5O 

0145  9999 

0320  9995 

0494  9988 

0669  9978 

0843  9964 

1O 

51 

0148  9999 

0323  9995 

0497  9988 

0671  9977 

0845  9964 

9 

52 

0151  9999 

0326  9995 

0500  9987 

0674  9977 

0848  9964 

8 

53 

0154  9999 

0329  9995 

0503  9987 

0677  9977 

0851  9964 

7 

54 

0157  9999 

0332  9995 

0506  9987 

0680  9977 

0854  9963 

6 

55 

0160  9999 

0334  9994 

0509  9987 

0683  9977 

0857  9963 

5 

56 

0163  9999 

0337  9994 

0512  9987 

0686  9976 

0860  9963 

4 

57 

0166  9999 

0340  9994 

0515  9987 

06S9  9976 

0863  9963 

3 

58 

0169  9999 

0343  9994 

0518  9987 

0692  9976 

0866  9962 

2 

59 

0172  9999 

0346  9994 

0520  9986 

0695  9976 

0869  9962 

1 

60 

0175  9999 
cos   sin 

0349  9994 
cos   sin 

0523  9986 

0698  9976 

0872  9962 

O 

t 

89° 

88° 

87° 

86° 

86° 

» 

NATURAL   SINES   AND   COSINES. 


5° 

6° 

7° 

8° 

9° 

f 

Mil     COS 

sin   cos 

sin   cos 

sin   cos 

sin   COB 

0 

0872  9962 

1045  9945 

1219  9925 

1392  9903 

1564  9877 

«'»< 

1 

0874  9962 

1048  9945 

1222  9925 

1395  9902 

1567  9876 

59 

2 

0877  9461 

1051  9945 

1224  9925 

1397  9902 

1570  9876 

58 

3 

0880  9961 

1054  9944 

1227  9924 

1400  9901 

1573  9876 

57 

4 

0883  9961 

1057  9944 

1230  9924 

1403  9901 

1576  9875 

56 

5 

0886  9961 

1060  9944 

1233  9924 

1406  9901 

1579  9875 

55 

6 

0889  9960 

1063  9943 

1236  9923 

1409  9900 

1582  9874 

54 

7 

0892  9960 

1066  9943 

1239  9923 

1412  9900 

1584  9874 

53 

8 

0895  9960 

1068  9943 

1241  9923 

1415  9899 

1587  9873 

52 

9 

0898  9960 

1071  9942. 

1245  9922 

1418  9899 

1590  9S73 

51 

10 

0901  9959 

1074  9942 

1248  9922 

1421  9899 

1593  9872 

5O 

11 

0903  9959 

1077  9942 

1250  9922 

1423  9898 

1596  9872 

49 

12 

0906  9959 

1080  9942 

1253  9921 

1426  9898 

1599  9871 

48 

13 

0909  9959 

1083  9941 

1256  9921 

1429  9897 

1602  9871 

47 

14 

0912  9958 

1086  9941 

1259  9920 

1432  9897 

1605  9870 

46 

15 

09]  5  9958 

1089  9941 

1262  9920 

1435  9897 

1607  9870 

45 

16 

0918  9958 

1092  9940 

1265  9920 

1438  9896 

1610  9869 

44 

17 

0921  9958 

1094  9940 

1268  9919 

1441  9896 

1613  9869 

43 

18 

0924  9957 

1097  9940 

1271  9919 

1444  9895 

1616  9869 

42 

19 

0927  9957 

1100  9939 

1274  9919 

1446  9895 

1619  9868 

41 

20 

0929  9957 

1103  9939 

1276  9918 

1449  9894 

1622  9868 

4O 

21 

0932  9956 

1106  9939 

1279  9918 

1452  9894 

1625  9867 

39 

22 

0935  9956 

1109  9938 

1282  9917 

1455  9894 

1628  9867 

38 

23 

0938  9956 

1112  9938 

1285  9917 

1458  9893 

1630  9866 

37 

24 

0941  9956 

1115  9938 

1288  9917 

1461  9893 

1633  9866 

36 

25 

0944  9955 

1118  9937 

1291  9916 

1464  9892 

1636  9865 

35 

26 

0947  9955 

1120  9937 

1294  9916 

1467  9892 

1639  9865 

34 

27 

0950  9955 

1123  9937 

1297  9916 

1469  9891 

1642  9864 

33 

28 

0953  9955 

1126  9936 

1299  9915 

1472  9891 

1645  9864 

32 

29 

0956  9954 

1129  9936 

1302  9915 

1475  9891 

1648  9863 

31 

30 

0958  9954 

1132  9936 

1305  9914 

1478  9890 

1650  9863 

3O 

31 

0961  9954 

1135  9935 

1308  9914 

1481  9890 

1653  9862 

29 

32 

9964  9953 

1138  9935 

1311  9914 

1484  9889 

1656  9862 

28 

33 

0967  9953 

1141  9935 

1314  9913 

1487  9889 

1659  9861 

27 

34 

0970  9953 

1144  9934 

1317  9913 

1490  9888 

1662  9861 

26 

35 

0973  9553 

1146  9934 

1320  9913 

1492  9888 

1665  9860 

25 

36 

0976  9952 

1149  9934 

1323  9912 

1495  9888 

1668  9860 

24 

37 

0979  9952 

1152  9933 

1325  9912 

1498  9887 

1671  9859 

23 

38 

0982  9952 

1155  9933 

1328  9911 

1501  9887 

1673  9859 

22 

39 

0985  9951 

1158  9933 

1331  9911 

1504  9886 

1676  9859 

21 

4O 

0987  9951 

1161  9932 

1334  9911 

1507  9886 

1679  9858 

2O 

41 

0990  9951 

1164  9932 

1337  9910 

1510  9885 

1682  9858 

19 

42 

0993  9951 

1167  9932 

1340  9910 

1513  9885 

1685  9857 

18 

43 

0996  9950 

1170  9931 

1343  9909 

1515  9884 

1688  9857 

17 

44 

0999  9950 

1172  9931 

1346  9909 

1518  9884 

1691  9856 

16 

45 

1002  9950 

1175  9931 

1349  9909 

1521  9884 

1693  9856 

15 

46 

1005  9949 

1178  9930 

1351  9908 

1524  9883 

1696  9855 

14 

47 

1008  9949 

1181  9930 

1354  9908 

1527  9883 

1699  9855 

13 

48 

1011  9949 

1184  9930 

1357  9907 

1530  9882 

1702  9854 

12 

49 

1013  9949 

1187  9929 

1360  9907 

1533  9882 

1705  9854 

11 

5O 

1016  9948 

1190  9929 

1363  9907 

1536  9881 

1708  9853 

1O 

51 

1019  9948 

1193  9929 

1366  9906 

1538  9881 

1711  9853 

9 

52 

1022  9948 

11%  9928 

1369  9906 

1541  9880 

1714  9852 

8 

53 

1025  9947 

1198  9928 

1372  9905 

1544  9880 

1716  9852 

7 

54 

1028  9947 

1201  9928 

1374  9905 

1547  9880 

1719  9851. 

6 

55 

1031  9947 

1204  9927 

1377  9905 

1550  9879 

1722  9851 

5 

56 

1034  9946 

1207  9927 

1380  9904 

1553  9879 

1725  9850 

4 

57 

1037  9946 

1210  9927 

1383  9904 

1556  9878 

1728  9850 

3 

58 

1039  9946 

1213  9926 

1386  9903 

1559  9878 

1731  9849 

2 

59 

1042  9946 

1216  9926 

1389  9903 

1561  9877 

1734  9849 

1 

60 

1045  9945 

1219  9925 

1392  9903 

1564  9877 

1736  9848 

0 

cos   sin 

cos   sin 

cos   ftin 

cos   sin 

COB    Bin 

f 

84° 

83° 

82° 

81° 

80° 

64 


NATURAL    SINES    AND    COSINES. 


f 

1O° 

11° 

12° 

13° 

14° 

r 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

o 

1736  9848 

1908  9816 

2079  9781 

2250  9744 

2419  9703 

6O 

1 

1739  9848 

1911  9816 

2082  9781 

2252  9743 

2422  9702 

59 

2 

1742  9847 

1914  9815 

2085  9780 

2255  9742 

2425  9702 

58 

3 

1745  9847 

1917  9815 

2088  9780 

2258  9742 

2428  9701 

57 

4 

1748  9846 

1920  9814 

2090  9779 

2261  9741 

2431  9700 

56 

5 

1751  9846 

1922  9813 

2093  9778 

2264  9740 

2433  9699 

55 

6 

1754  9845 

1925  9813 

2096  9778 

2267  9740 

2436  9699 

54 

7 

1757  9845 

1928  9812 

2099  9777 

2269  9739 

2439  9698 

53 

8 

1759  9844 

1931  9812 

2102  9777 

2272  9738 

2442  9697 

52 

9 

1762  9843 

1934  9811 

2105  9776 

2275  9738 

2445  9697 

51 

1O 

1765  9843 

1937  9811 

2108  9775 

2278  9737 

2447  9696 

50 

11 

1768  9842 

1939  9810 

2110  9775 

2281  9736 

2450  9695 

49 

12 

1771  9842 

1942  9810 

2113  9774 

2284  9736 

2453  9694 

48 

13 

1774  9841 

1945  9809 

2116  9774 

2286  9735 

2456  9694 

47 

14 

1777  9841 

1948  9808 

2119  9773 

2289  9734 

2459  9693 

46 

15 

1779  9840 

1951  9808 

2122  9772 

2292  9734 

2462  9692 

45 

16 

1782  9840 

1954  9S07 

2125  9772 

2295  9733 

2464  9692 

44 

17 

1785  9839 

1957  9807 

2127  9771 

2298  9732 

2467  9691 

43 

18 

1788  9839 

1959  9806 

2130  9770 

2300  9732 

2470  9690 

42 

19 

1791  9838 

1962  9806 

2133  9770 

2303  9731 

2473  9689 

41 

2O 

1794  9838 

1965  9805 

2136  9769 

2306  9730 

2476  9689 

4O 

21 

1797  9837 

1968  9804 

2139  9769 

2309  9730 

2478  9688 

39 

22 

1799  9837 

1971  9804 

2142  9768 

2312  9729 

2481  9687 

38 

23 

1802  9836 

1974  9803 

2145  9767 

2315  9728 

2484  9687 

37 

24 

1805  9836 

1977  9803 

2147  9767 

2317  9728 

2487  9686 

36 

25 

1808  9835 

1979  9802 

2150  9766 

2320  9727 

2490  9685 

35 

26 

1811  9835 

1982  9802 

2153  9765 

2323  9726 

2493  9684 

34 

27 

1814  9S34 

1985  9801 

2156  9765 

2326  9726 

2495  9684 

33 

28 

1817  9834 

1988  9800 

2159  9764 

2329  9725 

2498  9683 

32 

29 

1819  9833 

1991  9800 

2162  9764 

2332  9724 

2501  9682 

31 

30 

1822  9833 

1994  9799 

2164  9763 

2334  9724 

2504  9681 

30 

31 

1825  9832 

1997  9799 

2167  9762 

2337  9723 

2507  9681 

29 

32 

1828  9831 

1999  9798 

2170  9762 

2340  9722 

2509  9680 

28 

33 

1831  9831 

2002  9798 

2173  9761 

3343  9722 

2512  9679 

27 

34 

1834  9830 

2005  9797 

2176  9760 

2346  9721 

2515  9679 

26 

35 

1837  9830 

2008  9796 

2179  9760 

2349  9720 

2518  9678 

25 

36 

1840  9829 

2011  9796 

2181  9759 

2351  9720 

2521  9677 

24 

37 

1842  9829 

2014  9795 

2184  9759 

2354  9719 

2524  9676 

23 

38 

1845  9828 

2016  9795 

2187  9758 

2357  9718 

2526  9676 

22 

39 

1848  9828 

2019  9794 

2190  9757 

2360  9718 

2529  9675 

21 

40 

1851  9827 

2022  9793 

2193  9757 

2363  9717 

2532  9674 

20 

41 

1854  9827 

2025  9793 

2196  9756 

2366  9716 

2535  9673 

19 

42 

1857  9826 

2028  9792 

2198  9755 

2368  9715 

2538  9673 

18 

43 

1860  9826 

2031  9792 

2201  9755 

2371  9715 

2540  9672 

17 

44 

1862  9825 

2034  9791 

2204  9754 

2374  9714 

2543  9671 

16 

45 

1865  9825 

2036  9790 

2207  9753 

2377  9713 

2546  9670 

15 

46 

1868  9824 

2039  9790 

2210  9753 

2380  9713 

2549  9670 

14 

47 

1871  9823 

2042  9789 

2213  9752 

2383  9712 

2552  9669 

13 

48 

1474  9823 

2045  9789 

2215  9751 

2385  9711 

2554  9668 

12 

49 

1877  9822 

2048  9788 

2218  9751 

2388  9711 

2557  9667 

11 

50 

1880  9822 

2051  9787 

2221  9750 

2391  9710 

2560  9667 

10 

51 

1882  9821 

2054  9787 

2224  9750 

2394  9709 

2563  9666 

9 

52 

1885  9821 

2056  9786 

2227  9749 

2397  9709 

2566  9665 

8 

53 

1888  9820 

2059  9786 

2230  9748 

2399  9708 

2569  9665 

7 

54 

1891  9820 

2062  9785 

2233  9748 

2402  9707 

2571  9664 

6 

55 

1894  9819 

2065  9784 

2235  9747 

2405  9706 

2574  9663 

5 

56 

1897  9818 

2068  9784 

2238  9746 

2408  9706 

2577  9662 

4 

57 

1900  9818 

2071  9783 

2241  9746 

2411  9705 

2580  9662 

3 

58 

1902  9817 

2073  9783 

2244  9745 

2414  9704 

2583  9661 

2 

59 

1905  9817 

2076  9782 

2247  9744 

2416  9704 

2585  9660 

1 

60 

1908  9816 

2079  9781 

2250  9744 

2419  9703 

2588  9659 

0 

cos   sin 

cos    sill 

cos   sin 

cos   sin 

cos   sin 

f 

79° 

78° 

77° 

76° 

75° 

r 

NATURAL   SINES   AND   COSINES. 


65 


f 

15° 

16° 

17° 

18° 

19° 

t 

sin    cos 

Bin   cos 

sin   cos 

sin   cos 

sin   cos 

o 

25S8  9659 

2756  9613 

2924  9563 

3090  9511 

3256  9455 

6O 

1 

2591  9659 

2759  9612 

2926  9562 

3093  9510 

3258  9454 

59 

2 

2594  9658 

2762  9611 

2929  9561 

30%  9509 

3261  9453 

58 

3 

2597  9657 

2765  9610 

2932  9560 

3098  9508 

3264  9452 

57 

4 

2599  9656 

2768  9609 

2935  9560 

3101  9507 

3267  9451 

56 

5 

2602  9655 

2770  9609 

2938  9559 

3104  9506 

3269  9450 

55 

6 

2605  9655 

2773  9608 

2940  9558 

3107  9505 

3272  9449 

54 

7 

2608  9654 

2776  9607 

2943  9557 

3110  9504 

3275  9449 

53 

8 

2611  9653 

2779  9606 

2946  9556 

3112  9503 

3278  9448 

52 

9 

2613  9652 

2782  9605 

2949  9555 

3115  9502 

3280  9447 

51 

10 

2616  9652 

2784  9605 

2952  9555 

3118  9502 

3283  9446 

5O 

11 

2619  9651 

2787  9604 

2954  9554 

3121  9501 

3286  9445 

49 

12 

2622  9650 

2790  9603 

2957  9553 

3123  9500 

3289  9444 

48 

13 

2625  9649 

2793  9602 

2960  9552 

3126  9499 

3291  9443 

47 

14 

2628  9649 

2795  9601 

2963  9551 

3129  9498 

3294  9442 

46 

15 

2630  9648 

2798  9600 

2965  9550 

3132  9497 

3297  9441 

45 

16 

2633  9647 

2801  9600 

2968  9549 

3134  94% 

3300  9440 

44 

17 

2636  9646 

2804  9599 

2971  9548 

3137  9495 

3302  9439 

43 

18 

2639  9646 

2807  9598 

2974  9548 

3140  9494 

3305  9438 

42 

19 

2642  9645 

2809  9597 

2977  9547 

3143  9493 

3308  9437 

41 

2O 

2644  9644 

2812  95% 

2979  9546 

3145  9492 

3311  9436 

•JO 

21 

2647  9643 

2815  9596 

2982  9545 

3148  9492 

3313  9435 

39 

22 

2650  9642 

2818  9595 

2985  9544 

3151  9491 

3316  9434 

38 

23 

2653  9642 

2821  9594 

2988  9543 

3154  9490 

3319  9433 

37 

24 

2656  9641 

2823  9593 

2990  9542 

3156  9489 

3322  9432 

36 

25 

2658  9640 

2826  9592 

2993  9542 

3159  9488 

3324  9431 

35 

26 

2661  9639 

2829  9591 

29%  9541 

3162  9487 

3327  9430 

34 

27 

2664  9639 

2832  9591 

2999  9540 

3165  9486 

3330  9429 

33 

28 

2667  9638 

2835  9590 

3002  9539 

3168  9485 

3333  9428 

32 

29 

2670  9637 

2837  9589 

3004  9538 

3170  9484 

3335  9427 

31 

30 

2672  9636 

2840  9588 

3007  9537 

3173  9483 

3338  9426 

3O 

31 

2675  9636 

2843  9587 

3010  9536 

3176  9482 

3341  9425 

29 

32 

2678  9635 

2846  9587 

3013  9535 

3179  9481 

3344  9424 

28 

33 

2681  9634 

2849  9586 

3015  9535 

3181  9480 

3346  9423 

27 

34 

2684  9633 

2851  9585 

3018  9534 

3184  9480 

3349  9423 

26 

35 

2686  9632 

2854  9584 

3021  9533 

3187  9479 

3352  9422 

25 

36 

2689  9632 

2857  9583 

3024  9532 

3190  9478 

3355  9421 

24 

37 

2692  9631 

2860  9582 

3026  9531 

3192  9477 

3357  9420 

23 

38 

2695  9630 

2862  9582 

3029  9530 

3195  9476 

3360  9419 

22 

39 

2698  9629 

2865  9581 

3032  9529 

3198  9475 

3363  9418 

21 

l«» 

2700  9628 

2868  9580 

3035  9528 

3201  9474 

3365  9417 

20 

41 

2703  9628 

2871  9579 

3038  9527 

3203  9473 

3368  9416 

19 

42 

2706  9627 

2874  9578 

3040  9527 

3206  9472 

3371  9415 

18 

43 

2709  9626 

2876  9577 

3043  9526 

3209  9471 

3374  9414 

17 

44 

2712  9625 

2879  9577 

3046  9525 

3212  9470 

3376  9413 

16 

45 

2714  9625 

2882  9576 

3049  9524 

3214  9469 

3379  9412 

15 

46 

2717  9624 

2885  9575 

3051  9523 

3217  9468 

3382  9411 

14 

47 

2720  9623 

2888  9574 

3054  9522 

3220  9467 

3385  9410 

13 

48 

2723  9622 

2890  9573 

3057  9521 

3223  9466 

3387  9409 

12 

49 

2726  9621 

2893  9572 

3060  9520 

3225  9466 

3390  9408 

11 

5O 

2728  9621 

2896  9572 

3062  9520 

3228  9465 

3393  9407 

1O 

51 

2731  9620 

2899  9571 

3065  9519 

3231  9464 

33%  9406 

9 

52 

2734  9619 

2901  9570 

3068  9518 

3234  9463 

3398  9405 

8 

53 

2737  9618 

2904  9569 

3071  9517 

3236  9462 

3401  9404 

7 

54 

2740  9617 

2907  9568 

3074  9516 

3239  9461 

3404  9403 

6 

55 

2742  9617 

2910  9567 

3076  9515 

3242  9460 

3407  9402 

5 

56 

2745  9616 

2913  9566 

3079  9514 

3245  9459 

3409  9401 

4 

57 

2748  9615 

2915  9566 

3082  9513 

3247  9458 

3412  9400 

3 

58 

2751  9614 

2918  9565 

3085  9512 

3250  9457 

3415  9399 

2 

59 

2754  9613 

2921  9564 

30S7  9511 

3253  9456 

3417  9398 

1 

60 

2756  9613 

2924  9563 

3090  9511 

3256  9455 

3420  9397 

0 

cos   Bin 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

/ 

74o 

73° 

72° 

71° 

7O°     ' 

66  NATURAL   SINES   AND   COSINES. 


t 

2O° 

21° 

22° 

23° 

24° 

f 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

o 

3420  9397 

3584  9336 

3746  9272 

3907  9205 

4067  9135 

6O 

1 

3423  9396 

3586  9335 

3749  9271 

3910  9204 

4070  9134 

59 

2 

3426  9395 

3589  9334 

3751  9270 

3913  9203 

4073  9133 

58 

3 

3428  9394 

3592  9333 

3754  9269 

3915  9202 

4075  9132 

57 

4 

3431  9393 

3595  9332 

3757  9267 

3918  9200 

4078  9131 

56 

5 

3434  9392 

3597  9331 

3760  9266 

3921  9199 

4081  9130 

55 

6 

3437  9391 

3600  9330 

3762  9265 

3923  9198 

4083  9128  |  54 

7 

3439  9390 

3603  9328 

3765  9264 

3926  9197 

4086  9127 

53 

8 

3442  9389 

3605  9327 

3768  9263 

3929  9196 

4089  9126 

52 

9 

3445  9388 

3608  9326 

3770  9262 

3931  9195 

4091  9125 

51 

1O 

3448  9387 

3611  9325 

3773  9261 

3934  9194 

4094  9124 

5O 

11 

3450  9386 

3614  9324 

3776  9260 

3937  9192 

4097  9122 

49 

12 

3453  9385 

3616  9323 

3778  9259 

3939  9191 

4099  9121 

48 

13 

3456  9384 

3619  9322 

3781  9258 

3942  9190 

4102  9120 

47 

14 

3458  9383 

3622  9321 

3784  9257 

3945  9189 

4105  9119 

46 

15 

3461  9382 

3624  9320 

3786  9255 

3947  9188 

4107  9118 

45 

16 

3464  9381 

3627  9319 

3789  9254 

3950  9187 

4110  9116 

44 

17 

3467  9380 

3630  9318 

3792  9253 

3953  9186 

4112  9115  i  43 

18 

3469  9379 

3633  9317 

3795  9252 

3955  9184 

4115  9114    42 

19 

3472  9378 

3635  9316 

3797  9251 

3958  9183 

4118  9113 

41 

2O 

3475  9377 

3638  9315 

3800  9250 

3961  9182 

4120  9112 

4O 

21 

3478  9376 

3641  9314 

3803  9249 

3963  9181 

4123  9110 

39 

22 

3480  9375 

3643  9313 

3805  9248 

3966  9180 

4126  9109 

38 

23 

3483  9374 

3646  9312 

3808  9247 

3969  9179 

4128  9108 

37 

24 

3486  9373 

3649  9311 

3811  9245 

3971  9178 

4131  9107 

36 

25 

3488  9372 

3651  9309 

3813  9244 

3974  9176 

4134  9106 

35 

26 

3491  9371 

3654  9308 

3816  9243 

3977  9175 

4136  9104 

34 

27 

3494  9370 

3657  9307 

3819  9242 

3979  9174 

4139  9103  !  33 

28 

3497  9369 

3660  9306 

3821  9241 

3982  9173 

4142  9102 

32 

29 

3499  9368 

3662  9305 

3824  9240 

3985  9172 

4144  9101 

31 

30 

3502  9367 

3665  9304 

3827  9239 

3987  9171 

4147  9100 

3O 

31 

3505  9366 

3668  9303 

3830  9238 

3990  9169 

4150  9098 

29 

32 

3508  9365 

3670  9302 

3832  9237 

3993  9168 

4152  9097 

28 

33 

3510  9364 

3673  9301 

3835  9235 

3995  9167 

4155  9096 

27 

34 

3513  9363 

3676  9300 

3838  9234 

3998  9166 

4158  9095 

26 

35 

3516  9362 

3679  9299 

3840  9233 

4001  9165 

4160  9094 

25 

36 

3518  9361 

3681  9298 

3843  9232 

4003  9164 

4163  9092 

24 

37 

3521  9360 

3684  9297 

3846  9231 

4006  9162 

4165  9091 

23 

38 

3524  9359 

3687  9296 

3848  9230 

4009  9161 

4168  9090 

22 

39 

3527  9358 

3689  9295 

3851  9229 

4011  9160 

4171  9089 

21 

40 

3529  9356 

3692  9293 

3854  9228 

4014  9159 

4173  9088 

2O 

41 

3532  9355 

3695  9292 

3856  9227 

4017  9158 

4176  9086 

19 

42 

3535  9354 

3697  9291 

3859  9225 

4019  9157 

4179  9085 

18 

43 

3537  9353 

3700  9290 

3862  9224 

4022  9155 

4181  9084 

17 

44 

3540  9352 

3703  9289 

3864  9223 

4025  9154 

4184  9083 

16 

45 

3543  9351 

3706  9288 

3867  9222 

4027  9153 

4187  9081 

15 

46 

3546  9350 

3708  9287 

3870  9221 

4030  9152 

4189  9080 

14 

47 

3548  9349 

3711  9286 

3872  9220 

4033  9151 

4192  9079 

13 

48 

3551  9348 

3714  9285 

3875  9219 

4035  9150 

4195  9078 

12 

49 

3554  9347 

3716  9284 

3878  9218 

4038  9148 

4197  9077 

11 

50 

3557  9346 

3719  9283 

3881  9216 

4041  9147 

4200  9075 

10 

51 

3559  9345 

3722  9282 

3883  9215 

4043  9146 

4202  9074 

9 

52 

3562  9344 

3724  9281 

3886  9214 

4046  9145 

4205  9073 

8 

53 

3565  9343 

3727  9279 

3889  9213 

4049  9144 

4208  9072 

7 

54 

3567  9342 

3730  9278 

3891  9212 

4051  9143 

4210  9070 

6 

55 

3570  9341 

3733  9277 

3894  9211 

4054  9141 

4213  9069 

5 

56 

3573  9340 

3735  9276 

3897  9210 

4057  9140 

4216  9068 

4 

57 

3576  9339 

3738  9275 

3899  9208 

4059  9139 

4218  9067 

3 

58 

3578  9338 

3741  9274 

3902  9207 

4062  9138 

4221  9066 

2 

59 

3581  9337 

3743  9273 

3905  9206 

4065  9137 

4224  9064 

1 

6O 

3584  9336 

3746  9272 

3907  9205 

4067  9135 

4226  9063 

O 

cos   gin 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

r 

69° 

68° 

67° 

66° 

65°      ' 

NATURAL   SINES   AND   COSINES. 


25° 

26° 

27° 

28° 

2i> 

t 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

sin   cog 

o 

4226  9063 

4384  8988 

4540  8910 

4695  8829 

4848  8746 

6O 

1 

4229  9062 

4386  8987 

4542  8909 

4697  8828 

4851  8745 

59 

2 

4231  9061 

4389  8985 

4545  8907 

4700  8827 

4853  8743 

58 

3 

4234  9059 

4392  8984 

4548  8906 

4702  8825 

4856  8742 

57 

4 

4237  9058 

4394  8983 

4550  8905 

4705  8824 

4858  8741 

56 

5 

4239  9057 

4397  8982 

4553  8903 

4708  8823 

4861  8739 

55 

6 

4242  9056 

4399  8980 

4555  8902 

4710  8821 

4863  8738 

54 

7 

4245  9054 

4402  8979 

4558  8901 

47J  3  8820 

4866  8736 

53 

8 

4247  9053 

4405  8978 

4561  8899 

4715  8819 

4868  8735 

52 

9 

4250  9052 

4407  8976 

4563  8898 

4718  8817 

4871  8733 

51 

10 

4253  9051 

4410  8975 

4566  8897 

4720  8816 

4874  8732 

5O 

11 

4255  9050 

4412  8974 

4568  8895 

4723  8814 

4876  8731 

49 

12 

4258  9048 

4415  8973 

4571  8894 

4726  8813 

4879  8729 

48 

13 

4260  9047 

4418  8971 

4574  8893 

4728  8812 

4881  8728 

47 

14 

4263  9046 

4420  8970 

4576  8892 

4731  8810 

4884  8726 

46 

15 

4266  9045 

4423  8969 

4579  8890 

4733  8809 

4886  8725 

45 

16 

4268  9043 

4425  8967 

4581  8889 

4736  8808 

4889  8724 

44 

17 

4271  9042 

4428  8966 

4584  8888 

4738  8806 

4891  8722 

43 

18 

4274  9041 

4431  S965 

4586  8886 

4741  8805 

4894  8721 

42 

19 

4276  9040 

4433  8964 

4589  8885 

4743  8803 

4896  8719 

41 

20 

4279  9038 

4436  S962 

4592  8884 

4746  8S02 

4899  8718 

40 

21 

4281  9037 

4439  8961 

4594  8882 

4749  8801 

4901  8716 

39 

22 

4284  9036 

4441  8960 

4597  8881 

4751  8799 

4904  8715 

38 

23 

4287  9035 

4444  8958 

4599  8879 

4754  8798 

4907  8714 

37 

24 

4289  9033 

4446  8957 

4602  8878 

4756  8796 

4909  8712 

36 

25 

4292  9032 

4449  8956 

4605  8877 

4759  8795 

4912  8711 

35 

26 

4295  9031 

4452  8955 

4607  8875 

4761  8794 

4914  8709 

34 

27 

4297  9030 

4454  8953 

4610  8874 

4764  8792 

4917  8708 

33 

28 

4300  9028 

4457  8952 

4612  8873 

4766  8791 

4919  8706 

32 

29 

4302  9027 

4459  8951 

4615  8871 

4769  8790 

4922  8705 

31 

3O 

4305  9026 

4462  8949 

4617  8870 

4772  8788 

4924  8704 

30 

31 

4308  9025 

4465  8948 

4620  8869 

4774  8787 

4927  8702 

29 

32 

4310  9023 

4467  8947 

4623  8867 

4777  8785 

4929  8701 

28 

33 

4313  9022 

4470  8945 

4625  8866 

4779  8784 

4932  8699 

27 

34 

4316  9021 

4472  8944 

4628  8865 

4782  8783 

4934  8698 

26 

35 

4318  9020 

4475  8943 

4630  8863 

4784  8781 

4937  8696 

25 

36 

4321  9018 

4478  8942 

4633  8862 

4787  8780 

4939  8695 

24 

37 

4323  9017 

4480  8940 

4636  8861 

4789  8778 

4942  8694 

23 

38 

4326  9016 

4483  8939 

4638  8859 

4792  8777 

4944  8692 

22 

39 

4329  9015 

4485  8938 

4641  8858 

4795  8776 

4947  8691 

21 

4O 

4331  9013 

4488  8936 

4643  8857 

4797  8774 

4950  8689 

2O 

41 

4334  9012 

4491  8935 

4646  8855 

4800  8773 

4952  8688 

19 

42 

4337  9011 

4493  8934 

4648  8854 

4802  8771 

4955  8686 

18 

43 

4339  9010 

4496  8932 

4651  8853 

4805  8770 

4957  8685 

17 

44 

4342  9008 

4498  8931 

4654  8851 

4807  8769 

4960  8683 

16 

45 

4344  9007 

4501  8930 

4656  8850 

4810  8767 

4962  8682 

15 

46 

4347  9006 

4504  8928 

4659  8849 

4812  8766 

4965  8681 

14 

47 

4350  9004 

4506  8927 

4661  8847 

4815  8764 

4967  8679 

13 

48 

4352  9003 

4509  8926 

4664  8846 

4818  8763 

4970  8678 

12 

49 

4355  9002 

4511  8925 

4666  8844 

4820  8762 

4972  8676 

11 

50 

4358  9001 

4514  8923 

4669  8843 

4823  8760 

4975  8675 

1O 

51 

4360  8999 

4517  8922 

4672  8842 

4825  8759 

4977  8673 

9 

52 

4363  8998 

4519  8921 

4674  8840 

4828  8757 

4980  8672 

8 

53 

4365  8997 

4522  8919 

4677  8839 

4830  8756 

4982  8670 

7 

54 

4368  8996 

4524  8918 

4679  8838 

4833  8755 

4985  8669 

6 

55 

4371  8994 

4527  8917 

4682  8836 

4835  8753 

4987  8668 

5 

56 

4373  8993 

4530  8915 

4684  8835 

4838  8752 

4990  8666 

4 

57 

4376  8992 

4532  8914 

4687  8834 

4840  8750 

4992  8665 

3 

58 

4378  8990 

4535  8913 

4690  8832 

4843  8749 

4995  8663 

2 

59 

4381  8989 

4537  8911 

4692  8831 

4846  8748 

4997  8662 

1 

60 

4384  8988 

4540  8910 

4695  8829 

4848  8746 

5000  8660 

0 

cos    sin 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

f 

64° 

63° 

62° 

01° 

OO° 

t 

68 


NATURAL   SINES   AND   COSINES. 


r 

3O° 

31° 

32° 

33° 

34° 

t 

sin   cos 

sin   cos 

sin   cos 

•in   cos 

sin   cos 

O 

5000  8660 

5150  8572 

5299  8480 

5446  8387 

5592  8290 

6O 

1 

5003  8659 

5153  8570 

5302  8479 

5449  8385 

5594  8289 

59 

2 

5005  8657 

5155  8569 

5304  8477 

5451  8384 

5597  8287 

58 

3 

5008  8656 

5158  8567 

5307  8476 

5454  8382 

5599  8285 

57 

4 

5010  8654 

5160  8566 

5309  8474 

5456  8380 

5602  8284 

56 

5 

5013  8653 

5163  8564 

5312  8473 

5459  8379 

5604  8282 

55 

6 

5015  8652 

5165  8563 

5314  8471 

5461  8377 

5606  8281 

54 

7 

5018  8650 

5168  8561 

5316  8470 

5463  8376 

5609  8279 

53 

8 

5020  8649 

5170  8560 

5319  8468 

5466  8374 

5611  8277 

52 

9 

5023  8647 

5173  8558 

5321  8467 

5468  8372 

5614  8276 

51 

1O 

5025  8646 

5175  8557 

5324  8465 

5471  8371 

5616  8274 

5O 

11 

5028  8644 

5178  8555 

5326  8463 

5473  8369 

5618  8272 

49 

12 

5030  8643 

5180  8554 

5329  8462 

5476  8368 

5621  8271 

48 

13 

5033  8641 

5183  8552 

5331  8460 

5478  8366 

5623  8269 

47 

14 

5035  8640 

5185  8551 

5334  8459 

5480  8364 

5626  8268 

46 

15 

5038  8638 

5188  8549 

5336  8457 

5483  8363 

5628  8266 

45 

16 

5040  8637 

5190  8548 

5339  8456 

5485  8361 

5630  8264 

44 

17 

5043  8635 

5193  8546 

5341  8454 

5488  8360 

5633  8263 

43 

18 

5045  8634 

5195  8545 

5344  8453 

5490  8358 

5635  8261 

42 

19 

5048  8632 

5198  8543 

5346  8451 

5493  8356 

5638  8259 

41 

2O 

5050  8631 

5200  8542 

5348  8450 

5495  8355 

5640  8258 

40 

21 

5053  8630 

5203  8540 

5351  8448 

5498  8353 

5642  8256 

39 

22 

5055  8628 

5205  8539 

5353  8446 

5500  8352 

5645  8254 

38 

23 

5058  8627 

5208  8537 

5356  8445 

5502  8350 

5647  8253 

37 

24 

5060  8625 

5210  8536 

5358  8443 

5505  8348 

5650  8251 

36 

25 

5063  8624 

5213  8534 

5361  8442 

5507  8347 

5652  8249 

35 

26 

5065  8622 

5215  8532 

5363  8440 

5510  8345 

5654  8248 

34 

27 

5068  8621 

5218  8531 

5366  8439 

5512  8344 

5657  8246 

33 

28 

5070  8619 

5220  8529 

5368  8437 

5515  8342 

5659  8245 

32 

29 

5073  8618 

5223  8528 

5371  8435 

5517  8340 

5662  8243 

31 

30 

5075  8616 

5225  8526 

5373  8434 

5519  8339 

5664  8241 

3O 

31 

5078  8615 

5227  8525 

5375  8432 

5522  8337 

5666  8240 

29 

32 

5080  8613 

5230  8523 

5378  8431 

5524  8336 

5669  8238 

28 

33 

5083  8612 

5232  8522 

5380  8429 

5527  8334 

5671  8236 

27 

34 

5085  8610 

5235  8520 

5383  8428 

5529  8332 

5674  8235 

26 

35 

5088  8609 

5237  8519 

5385  8426 

5531  8331 

5676  8233 

25 

36 

5090  8607 

5240  8517 

5388  8425 

5534  8329 

5678  8231 

24 

37 

5093  8606 

5242  8516 

5390  8423 

5536  8328 

5681  8230 

23 

38 

5095  8604 

5245  8514 

5393  8421 

5539  8326 

5683  8228 

22 

39 

5098  8603 

5247  8513 

5395  8*20 

5541  8324 

5686  8226 

21 

40 

5100  8601 

5250  8511 

5398  8418 

5544  8323 

5688  8225 

2O 

41 

5103  8600 

5252  8510 

5400  8417 

5546  8321 

5690  8223 

19 

42 

5105  8599 

5255  8508 

5402  8415 

5548  8320 

5693  8221 

18 

43 

5108  8597 

5257  8507 

5405  8414 

5551  8318 

5695  8220 

17 

44 

5110  8596 

5260  8505 

5407  8412 

5553  8316 

5698  8218 

16 

45 

S3  13  8594 

5262  8504 

5410  8410 

5556  8315 

5700  8216 

15 

46 

5115  8i93 

5265  8502 

5412  8409 

5558  8313 

5702  8215 

14 

47 

.5118  8591 

5267  8500 

5415  8407 

5561  8311 

5705  8213 

13 

48 

5120  8590 

5270  8499 

5417  8406 

5563  8310 

5707  8211 

12 

49 

5123  8588 

5272  8497 

5420  8404 

5565  8308 

5710  8210 

11 

50 

5125  8587 

5275  84% 

5422  8403 

5568  8307 

5712  8208 

10 

51 

5128  8585 

5277  8494 

5424  8401 

5570  8305 

5714  8207 

9 

52 

5130  8584 

5279  8493 

5427  8399 

5573  8303 

5717  8205 

8 

53 

5133  8582 

5282  8491 

5429  8398 

5575  8302 

5719  8203 

7 

54 

5135  8581 

5284  8490 

5432  83% 

5577  8300 

5721  8202 

6 

55 

5138  8579 

5287  8488 

5434  8395 

5580  8299 

5724  8200 

5 

56 

5140  8578 

5289  8487 

5437  8393 

5582  8297 

5726  8198 

4 

57 

5143  8576 

5292  8485 

5439  8391 

5585  8295 

5729  8197 

3 

58 

5145  8575 

5294  8484 

5442  8390 

5587  8294 

5731  8195 

2 

59 

5148  8573 

5297  8482 

5444  8388 

5590  8292 

5733  8193 

1 

60 

5150  8572 

5299  8480 

5446  8387 

5592  8290 

5736  8192 

0 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

t 

59° 

58° 

57° 

56° 

55° 

t 

NATURAL  SINES   AND   COSINES.  69 


t 

35° 

36° 

37° 

38° 

39°      , 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

gin   cos 

o 

5736  8192 

5878  8090 

6018  7986 

6157  7880 

6293  7771 

<•><> 

1 

5738  8190 

5880  8088 

6020  7985 

6159  7878 

6295  7770 

59 

2 

5741  8188 

58S3  8087 

6023  7983 

6161  7877 

6298  7768 

58 

3 

5743  8187 

5885  8085 

6025  7981 

6163  7875 

6300  7766 

57 

4 

5745  8185 

5887  8083 

6027  7979 

6166  7873 

6302  7764 

56 

5 

5748  8183 

5890  8082 

6030  7978 

6168  7871 

6305  7762 

55 

6 

5750  8181 

5892  8080 

6032  7976 

6170  7869 

6307  7760 

54 

7 

5752  8180 

5894  8078 

6034  7974 

6173  7868 

6309  7759 

53 

8 

5755  8178 

5897  8076 

6037  7972 

6175  7866 

6311  7757 

52 

9 

5757  8176 

5899  8075 

6039  7971 

6177  7864 

6314  7755 

51 

1O 

5760  8175 

5901  8073 

6041  7969 

6180  7862 

6316  7753 

5O 

11 

5762  8173 

5904  8071 

6044  7967 

6182  7860 

6318  7751 

49 

12 

5764  8171 

5906  8070 

6046  7965 

6184  7859 

6320  "7749 

48 

13 

5767  8170 

5908  8068 

6048  7964 

6186  7857 

6323  7748 

47 

14 

5769  8168 

5911  8066 

6051  7962 

6189  7855 

6325  7746 

46 

15 

5771  8166 

5913  8064 

6053  7960 

6191  7853 

6327  7744 

45 

16 

5774  8165 

5915  8063 

6055  7958 

6193  7851 

6329  7742 

44 

17 

5776  8163 

5918  8061 

6058  7956 

6196  7850 

6332  7740 

43 

18 

5779  8161 

5920  8059 

6060  7955 

6198  7848 

6334  7738 

42 

19 

5781  8160 

5922  8058 

6062  7953 

6200  7846 

6336  7737 

41 

2O 

5783  8158 

5925  8056 

6065  7951 

6202  7844 

6338  7735 

40 

21 

5786  8156 

5927  8054 

6067  7950 

6205  7842 

6341  7733 

39 

22 

5788  8155 

5930  8052 

6069  7948 

6207  7841 

6343  7731 

38 

23 

5790  8153 

5932  8051 

6071  7946 

6209  7839 

6345  7729 

37 

24 

5793  8151 

5934  8049 

6074  7944 

6211  7837 

6347  7727 

36 

25 

5795  8150 

5937  8047 

6076  7942 

6214  7835 

6350  7725 

35 

26 

5798  8148 

5939  8045 

6078  7941 

6216  7833 

6352  7724 

34 

27 

5800  8146 

5941  8044 

6081  7939 

6218  7832 

6354  7722 

33 

28 

5802  8145 

5944  8042 

6083  7937 

6221  7830 

6356  7720 

32 

29 

5805  8143 

5946  8040 

6085  7935 

6223  7828 

6359  7718 

31 

30 

5807  8141 

5948  8039 

6088  7934 

6225  7826 

6361  7716 

3O 

31 

5809  8139 

5951  8037 

6090  7932 

6227  7824 

6363  7714 

29 

32 

5812  S138 

5953  8035 

6092  7930 

6230  7822 

6365  7713 

28 

33 

5814  8136 

5955  8033 

6095  7928 

6232  7821 

6368  7711 

27 

34 

5816  8134 

5958  8032 

6097  7926 

6234  7819 

6370  7709 

26 

35 

5819  8133 

5960  8030 

6099  7925 

6237  7817 

6372  7707 

25 

36 

5821  8131 

5962  S02S 

6101  7923 

6239  781  5 

6374  7705 

24 

37 

5824  8129 

5965  8026 

6104  7921 

6241  7813 

6376  7703 

23 

38 

5826  8128 

5967  8025 

6106  7919 

6243  7812 

6379  7701 

22 

39 

5828  8126 

5969  8023 

6108  7918 

6246  7810 

6381  7700 

21 

4O 

5831  8124 

5972  8021 

6111  7916 

6248  7808 

6383  7698 

20 

41 

5833  8123 

5974  8020 

6113  7914 

6250  7S06 

6385  7696 

19 

42 

5835  8121 

5976  8018 

6115  7912 

6252  7804 

6388  7694 

18 

43 

583S  8119 

5979  8016 

6118  7910 

6255  7802 

6390  7692 

17 

44 

5840  8117 

5981  8014 

6120  7909 

6257  7801 

6392  7690 

16 

45 

5842  8116 

5983  8013 

6122  7907 

6259  7799 

6394  7688 

15 

46 

5S45  8114 

5986  8011 

6124  7905 

6262  7797 

6397  7687 

14 

47 

5847  8112 

5988  8009 

6127  7903 

6264  7795 

6399  7685 

13 

48 

5850  8111 

5990  8007 

6129  7902 

6266  7793 

6401  7683 

12 

49 

5852  8109 

5993  8006 

6131  7900 

6268  7792 

6403  7681 

11 

5O 

5854  8107 

5995  8004 

6134  7898 

6271  7790 

6406  7679 

10 

51 

5857  8106 

5997  8002 

6136  7896 

6273  7788 

6408  7677 

9 

52 

5859  8104 

6000  8000 

6138  7894 

6275  7786 

6410  7675 

8 

53 

5861  8102 

6002  7999 

6141  7893 

6277  7784 

6412  7674 

7 

54 

5864  8100 

6004  7997 

6143  7891 

6280  7782 

6414  7672 

6 

55 

5866  8099 

6007  7995 

6145  7889 

6282  7781 

6417  7670 

5 

56 

5868  8097 

6009  7993 

6147  7887 

6284  7779 

6419  7668 

4 

57 

587  1  8095 

6011  7992 

6150  7885 

6286  7777 

6421  7666 

3 

58 

5873  8094 

6014  7990 

6152  7884 

6289  7775 

6423  7664 

2 

59 

5875  8092 

6016  7988 

6154  7882 

6291  7773 

6426  7662 

1 

(JO 

5S7S  8090 

6018  7986 

6157  7880 

6293  7771 

6428  7660 

0 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

COS    Hill 

'      54° 

r>:$c 

r>2^ 

5T" 

50°      ' 

70  NATURAL   SINES   AND   COSINES. 


r 

40° 

41° 

42° 

43° 

44° 

t 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

sin   cos 

O 

6428  7660 

6561  7547 

6691  7431 

6820  7314 

6947  7193 

6O 

1 

6430  7659 

6563  7545 

6693  7430 

6822  7312 

6949  7191 

59 

2 

6432  7657 

6565  7543 

6696  7428 

6824  7310 

6951  7189 

58 

3 

6435  7655 

6567  7541 

6698  7426 

6826  7308 

6953  7187 

57 

4 

6437  7653 

6569  7539 

6700  7424 

6828  7306 

6955  7185 

56 

5 

6439  7651 

6572  7538 

6702  7422 

6831  7304 

6957  7183 

55 

6 

6441  7649 

6574  7536 

6704  7420 

6833  7302 

6959  7181    54 

7 

6443  7647 

6576  7534 

6706  7418 

6835  7300 

6961  7179 

53 

8 

6446  7645 

6578  7532 

6709  7416 

6837  7298 

6963  7177 

52 

9 

6448  7644 

6580  7530 

6711  7414 

6839  7296 

6965  7175 

51 

1O 

6450  7642 

6583  7528 

6713  7412 

6841  7294 

6967  7173 

50 

11 

6452  7640 

6585  7526 

6715  7410 

6843  7292 

6970  7171 

49 

12 

6455  7638 

6587  7524 

6717  7408 

6845  7290 

6972  7169 

48 

13 

6457  7636 

6589  7522 

6719  7406 

6848  7288 

6974  7167 

47 

14 

6459  7634 

6591  7520 

6722  7404 

6850  7286 

6976  7165 

46 

15 

6461  7632 

6593  7518 

6724  7402 

6852  7284 

6978  7163 

45 

16 

"6463  7630 

6596  7516 

6726  7400 

6854  7282 

6980  7161 

44 

17 

6466  7629 

6598  7515 

6728  7398 

6856  7280 

6982  7159 

43 

18 

6468  7627 

6600  7513 

6730  7396 

6858  7278 

6984  7157 

42 

19 

6470  7625 

6602  7511 

6732  7394 

6S60  7276 

6986  7155 

41 

2O 

6472  7623 

6604  7509 

6734  7392 

6862  7274 

6988  7153 

40 

21 

6475  7621 

6607  7507 

6737  7390 

6865  7272 

6990  7151 

39 

22 

6477  7619 

6609  7505 

6739  7388 

6867  7270 

6992  7149 

38 

23 

6479  7617 

6611  7503 

6741  7387 

6869  7268 

6995  7147 

37 

24 

6481  7615 

6613  7501 

6743  7385 

6871  7266 

6997  7145 

36 

25 

6483  7613 

6615  7499 

6745  7383 

6873  7264 

6999  7143 

35 

26 

6486  7612 

6617  7497 

6747  7381 

6875  7262 

7001  7141 

34 

27 

6488  7610 

6620  7495 

6749  7379 

6877  7260 

7003  7139 

33 

28 

6490  7608 

6622  7493 

6752  7377 

6879  7258 

7005  7137 

32 

29 

6492  7606 

6624  7491 

6754  7375 

6881  7256 

7007  7135 

31 

30 

6494  7604 

6626  7490 

6756  7373 

6884  7254 

7009  7133 

3O 

31 

6497  7602 

6628  7488 

6758  7371 

6886  7252 

7011  7130 

29 

32 

6499  7600 

6631  7486 

6760  7369 

6888  7250 

7013  7128 

28 

33 

6501  7598 

6633  7484 

6762  7367 

6890  7248 

7015  7126 

27 

34 

6503  7596 

6635  7482 

6764  7365 

6892  7246 

7017  7124 

26 

35 

6506  7595 

6637  7480 

6767  7363 

6894  7244 

7019  7122 

25 

36 

6508  7593 

6639  7478 

6769  7361 

6896  7242 

7022  7120 

24 

37 

6510  7591 

6641  7476 

6771  7359 

6898  7240 

7024  7118 

23 

38 

6512  7589 

6644  7474 

6773  7357 

6900  7238 

7026  7116 

22 

39 

6514  7587 

6646  7472 

6775  7355 

6903  7236 

7028  7114 

21 

40 

6517  7585 

6648  7470 

6777  7353 

6905  7234 

7030  7112 

2O 

41 

6519  7583 

6650  7468 

6779  7351 

6907  7232 

7032  7110 

19 

42 

6521  7581 

6652  7466 

6782  7349 

6909  7230 

7034  7108 

18 

43 

6523  7579 

6654  7464 

6784  7347 

6911  7228 

7036  7106 

17 

44 

6525  7578 

6657  7463 

6786  7345 

6913  7226 

7038  7104 

16 

45 

6528  7576 

6659  7461 

6788  7343 

6915  7224 

7040  7102 

15 

46 

6530  7574 

6661  7459 

6790  7341 

6917  7222 

7042  7100 

14 

47 

6532  7572 

6663  7457 

6792  7339 

6919  7220 

7044  7098 

13 

48 

6534  7570 

6665  7455 

6794  7337 

6921  7218 

7046  7096 

12 

49 

6536  7568 

6667  7453 

6797  7335 

6924  7216 

7048  7094 

11 

50 

6539  7566 

6670  7451 

6799  7333 

6926  7214 

7050  7092 

10 

51 

6541  7564 

6672  7449 

6801  7331 

6928  7212 

7053  7090 

9 

52 

6543  7562 

6674  7447 

6803  7329 

6930  7210 

7055  7088 

8 

53 

6545  7560 

6676  7445 

6805  7327 

6932  7208 

7057  7085 

7 

54 

6547  7559 

6678  7443 

6807  7325 

6934  7206 

7059  7083 

6 

55 

6550  7557 

6680  7441 

6809  7323 

6936  7203 

7061  7081 

5 

56 

6552  7555 

6683  7439 

6811  7321 

6938  7201 

7063  7079 

4 

57 

6554  7553 

6685  7437 

6814  7319 

6940  7199 

7065  7077 

3 

58 

6556  7551 

6687  7435 

6816  7318 

6942  7197 

7067  7075 

2 

59 

6558  7549 

6689  7433 

6818  7316 

6944  7195 

7069  7073 

1 

6O 

6561  7547 

6691  7431 

6820  7314 

6947  7193 

7071  7071 

O 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

cos   sin 

f 

49° 

48° 

47° 

46° 

45° 

f 

TABLE  IX,— NATURAL  TANGENTS  AND  COTANGENTS.  ' 


2° 

3° 

4° 

r 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O  10000 

Infinite 

0175 

57.2900 

0349 

28.6363 

0524 

19.0811 

0699 

14.3007 

GO 

1   0003 

3437.75 

0177 

56.3506 

0352 

28.3994 

0527 

18.9755 

0702 

14.2411 

59 

2  i0006 

1718.87 

0180 

55.4415 

0355 

28.1664 

0530 

18.8711 

0705 

14.1821 

58 

3  10009 

1145.92 

0183 

54.5613 

0358 

27.9372 

0533 

18.7678 

0708 

14.1235 

57 

4  0012 

859.436 

0186 

53.7086 

0361 

27.7117 

0536 

18.6656 

0711 

14.0655 

56 

5   0015 

687.549 

0189 

52.8821 

0364 

27.4899 

0539 

18.5645 

0714 

14.0079 

55 

6  J0017 

572.957 

0192 

52.0807 

0367 

27.2715 

0542 

18.4645 

0717 

13.9507 

54 

7 

0020 

491.106 

0195 

51.3032 

0370 

27.0566 

0544 

18.3655 

0720 

13.8940 

53 

8  0023 

429.718 

0198 

50.5485 

0373 

26.8450 

0547 

18.2677 

0723 

13.8378 

52 

9  0026 

381.971 

0201 

49.8157 

0375 

26.6367 

0550 

18.1708 

0726 

13.7821 

51 

1O  0029 

343.774 

0204 

49.1039 

037S 

26.4316 

0553 

18.0750 

0729 

13.7267 

5O 

11  0032 

312.521 

0207 

48.4121 

0381 

26.2296 

0556 

17.9802 

0731 

13.6719 

49 

12  0035 

286.478 

0209 

47.7395 

0384 

26.0307 

0559 

17.8863 

0734 

13.6174 

48 

13 

0038 

264.441 

0212 

47.0853 

0387 

25.8348 

0562 

17.7934 

0737 

13.5634 

47 

14 

0041 

245.552 

0215 

46.4489 

0390 

25.6418 

0565 

17.7015 

0740 

13.5098  46 

15 

0044 

229.182 

0218 

45.8294 

0393 

25.4517 

0568 

17.6106 

0743 

13.4566  45 

16 

0047 

214.858 

0221 

45.2261 

0396 

25.2644 

0571 

17.5205 

0746 

13.4039!  44 

17 

0049 

202.219 

0224 

44.6386 

0399 

25.0798 

0574 

17.4314 

0749 

13.3515  43 

18 

0052 

190.984 

0227 

44.0661 

0402 

24.8978 

0577 

17.3432 

0752 

13.2996  42 

19 

0055 

180.932 

0230 

43.5081 

0405 

24.7185 

0580 

17.2558 

0755 

13.2480  .  41 

20 

0058 

171.885 

0233 

42.9641 

0407 

24.5418 

0582 

17.1693 

0758 

13.1%9  4O 

21 

0061 

163.700 

0236 

42.4335 

0410 

24.3675 

0585 

17.0837 

0761 

13.1461  39 

22 

0064 

156.259 

0239 

41.9158 

0413 

24.1957 

0588 

16.9990 

0764 

13.0958!  38 

23 

0067 

149.465 

0241 

41.4106 

0416 

24.0263 

0591 

16.9150 

0767 

13.0458  37 

24  0070 

143.237 

0244 

40.9174 

0419 

23.8593 

0594 

16.8319 

0769 

12.9962  36 

25  0073 

137.507 

0247 

40.4358 

0422 

23.6945 

0597 

16.74% 

0772 

12.9469  35 

26  0076 

132.219 

0250 

39.9655 

0425 

23.5321 

0600 

16.6681 

0775 

12.8981  j  34 

27  0079 

127.321 

0253 

39.5059 

0428 

23.3718 

0603 

16.5874 

0778 

12.84%  |  33 

28  i  OOS1 

122.774 

0256 

39.0568 

0431 

23.2137 

0606 

16.5075 

0781 

12.8014  32 

29 

0084 

118.540 

0259 

38.6177 

0434 

23.0577 

0609 

16.4283 

0784 

12.7536 

31 

3O 

0087 

114.589 

0262 

38.1885 

0437 

22.9038 

0612 

16.3499 

0787 

12.7062 

30 

31 

0090 

110.892 

0265 

37.7686 

0440 

22.7519 

0615 

16.2722 

0790 

12.6591  !  29 

32 

0093 

107.426 

0268 

37.3579 

0442 

22.6020 

0617 

16.1952 

0793 

12.6124  i  28 

33 

0096 

104.171 

0271 

36.9560 

0445 

22.4541 

0620 

16.1190 

07% 

12.5660  j  27 

34 

0099 

101.107 

0274 

36.5627 

0448 

22.3081 

0623 

16.0435 

0799 

12.5199!  26 

35 

0102 

98.2179 

0276 

36.1776 

0451 

22.1640 

0626 

15.9687 

0802 

12.4742  25 

36 

0105 

95.4895 

0279 

35.8006 

0454 

22.0217 

0629 

15.8945 

0805 

12.4288!  24 

37 

0108 

92.9085 

0282 

35.4313 

0457 

21.8813 

0632 

15.8211 

0808 

12.3838  23 

38 

0111 

90.4633 

0285 

35.0695 

0460 

21.7426 

0635 

15.7483 

0810 

12.3390  22 

39 

0113 

88.1436 

0288 

34.7151 

0463 

21.6056 

0638 

15.6762 

0813 

12.29461  21 

4O 

0116 

85.9398 

0291 

34.3678 

0466 

21.4704 

0641 

15.6048 

0816 

12.2505  2O 

41 

0119 

83.8435 

0294 

34.0273 

0469 

21.3369 

0644 

15.5340 

0819 

12.20671  19 

42  0122 

81.8470 

0297 

33.6935 

0472 

21.2049 

0647 

15.4638 

0822 

12.16321  18 

43  1Q12S 

79.9434 

0300 

33.3662 

0475 

21.0747 

0650 

15.3943 

0825 

12.1201  17 

44  0128 

78.1263 

0303 

33.0452 

0477 

20.9460 

0653 

15.3254 

0828 

12.0772  16 

45 

0131 

76.3900 

0306 

32.7303 

0480 

20.8188 

0655 

15.2571 

0831 

12.03461  IS 

46 

0134 

74.7292 

0308 

32.4213 

0483 

20.6932 

0658 

15.1893 

0834 

11.9923  14 

47 

0137 

73.1390 

0311 

32.1181 

0486 

20.5691 

0661 

15.1222 

0837 

11.9504 

13 

48 

0140 

71.6151 

0314 

31.8205 

0489 

20.4465 

0664 

15.0557 

0840 

11.9087  12 

49 

0143 

70.1533 

0317 

31.5284 

0492 

20.3253 

0667 

14.9898 

0843 

11.8673  11 

5O 

OJ46 

68.7501 

0320 

31.2416 

0495 

20.2056 

0670 

14.9244 

0846 

11.8262  1O 

51 

0148 

67.4019 

0323 

30.9599 

0498 

20.0872 

0673 

14.85% 

0849 

11.7853  9 

52  0151 

66.1055 

0326 

30.6833 

0501 

19.9702 

0676 

14.7954 

0851 

11.7448 

8 

53  0154 

64.8580 

0329 

30.4116 

0504 

19.8546 

0679 

14.7317 

0854 

11.7045 

7 

54 

0157 

63.6567 

0332 

30.1446 

0507 

19.7403 

0682 

14.6685 

0857 

11.6645 

6 

55 

0160 

62.4992 

0335 

29.8823 

0509 

19.6273 

0685 

14.6059 

0860 

11.6248 

5 

56 

0163 

61.3829 

0338 

29.6245 

0512 

19.5156 

0688 

14.5438 

0863 

11.5853  4 

57 

0166 

60.3058 

0340 

29.3711 

0515 

19.4051 

0690 

14^823 

0866 

11.5461 

3 

58 

0169 

59.2659 

0343 

29.1220 

0518 

19.2959 

0693 

14.4212 

0669 

11.5072 

2 

59 

0172 

58.2612 

0346 

28.8771 

0521 

19.1879 

0696 

14.3607 

0872 

11.4685 

1 

6O 

0175 

57.2900 

0349 

28.6363 

0524 

19.0811 

0699 

14.3007 

0875 

11.4301 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

'     89° 

88° 

87° 

80° 

S.T 

t 

72 


NATURAL   TANGENTS   AND   COTANGENTS. 


t 

5° 

6° 

7° 

8° 

9o       , 

tan    cot 

tan   cot 

tan   cot 

tan   cot 

tan    cot 

0 

0875  11.4301 

1051  9.5144 

1228  8.1443 

1405  7.1154 

1584  6.3138  !  GO 

1 

0878  11.3919 

1054  9.4878 

1231  8.1248 

1408  7.1004 

1587  6.3019  59 

2 

0881  11.3540 

1057  9.4614 

1234  8.1054 

1411  7.0855 

1590  6.2901  58 

3 

0884  11.3163 

1060  9.4352 

1237  8.0860 

1414  7.0706 

1593  6.2783  57 

4 

0887  11.2789 

1063  9.4090 

1240  8.0667 

1417  7.0558 

1596  6.2666  56 

5 

0890  11.2417 

1066  9.3831 

1243  8.0476 

1420  7.0410 

1599  6.2549  55 

6 

0892  11.2048 

1069  9.3572 

1246  8.0285 

1423  7.0264 

1602  6.2432  i  54 

7 

OS95  11.1681 

1072  9.3315 

1249  8.0095 

1426  7.0117 

1605  6.2316  53 

8 

0898  11.1316 

1075  9.3060 

1251  7.9906 

1429  6.9972 

1608  6.2200  j  52 

9 

0901  11.0954 

1078  9-2806 

1254  7.9718 

1432  6.9827 

1611  6.2085  51 

1O 

0904  11.0594 

1080  9.2553 

1257  7.9530 

1435  6.9682 

1614  6.1970  SO 

11 

0907  11.0237 

1083  9.2302 

1260  7.9344 

1438  6.9538 

1617  6.1856  49 

12 

0910  10.9882 

1086  9.2052 

1263  7.9158 

1441  6.9395 

1620  6.1742  48 

13 

0913  10.9529 

1089  9.1803 

1266  7.8973 

1444  6.9252 

1623  6.1628  47 

14 

0916  10.9178 

1092  9.1555 

1269  7.8789 

1447  6.9110 

1626  6.1515  46 

15 

0919  10.8829 

1095  9.1309 

1272  7.8606 

1450  6.8969 

1629  6.1402  45 

16 

0922  10.8483 

1098  9.1065 

1275  7.8424 

1453  6.8828 

1632  6.1290  44 

17 

0925  10.8139 

1101  9.0821 

1278  7.8243 

1456  6.8687 

1635  6.1178  43 

18 

0928  10.7797 

1104  9.0579 

1281  7.8062 

1459  6.8548 

1638  6.1066  i  42 

19 

0931  10.7457 

1107  9.0338 

1284  7.7883 

1462  6.8408 

1641  6.0955  i  41 

2O 

0934  10.7119 

1110  9.0098 

1287  7.7704 

1465  6.8269 

1644  6.0844  4O 

21 

0936  10.6783 

1113  8.9860 

1290  7.7525 

1468  6.8131 

1647  6.0734  39 

22 

0939  10.6450 

1116  8.9623 

1293  7.7348 

1471  6.7994 

1650  6.0624  38 

23 

0942  10.6118 

1119  8.9387 

1296  7.7171 

1474  6.7856 

1653  6.0514  !  37 

24 

0945  10.5789 

1122  8.9152 

1299  7.6996 

1477  6.7720 

1655  6.0405  36 

25 

0948  10.5462 

1125  8.8919 

1302  7.6821 

1480  6.7584 

1658  6.0296  |  35 

26 

0951  10.5136 

1128  8.8686 

1305  7.6647 

1483  6.7448 

1661  6.0188  34 

27 

0954  10.4813 

1131  8.8455 

1308  7.6473 

1486  6.7313 

1664  6.0080  '  33 

28 

0957  10.4491 

1134  8.8225 

1311  7.6301 

1489  6.7179 

1667  5.9972  ;  32 

29 

0960  10.4172 

1136  8.7996 

1314  7.6129 

1492  6.7045 

1670  5.9865  '  31 

3O 

0963  10.3854 

1139  8.7769 

1317  7.5958 

1495  6.6912 

1673  5.9758  3O 

31 

0966  10.3538 

1142  8.7542 

1319  7.5787 

1497  6.6779 

1676  5.9651  29 

32  10969  10.3224 

1145  8.7317 

1322  7.5618 

1500  6.6646 

1679  5.9545  28 

33  0972  10.2913 

1148  8.7093 

1325  7.5449 

1503  6.6514 

1682  5.9439  27 

34 

0975  10.2602 

1151  8.6870 

1328  7.5281 

1506  6.6383 

1685  5.9333  26 

35 

0978  10.2294 

1154  8.6648 

1331  7.5113 

1509  6.6252 

1688  5.9228  25 

36 

0981  10.1988 

1157  8.6427 

1334  7.4947 

1512  6.6122 

1691  5.9124  24 

37 

0983  10.1683 

1160  8.6208 

1337  7.4781 

1515  6.5992 

1694  5.9019  23 

38  J0986  10.1381 

1163  8.5989 

1340  7.4615 

1518  6.5863 

1697  5.8915  22 

39  0989  10.1080 

1166  8.5772 

1343  7.4451 

1521  6.5734 

1700  5.8811  21 

4O  0992  10.0780 

1169  8.5555 

1346  7.4287 

1524  6.5606 

1703  5.8708  2O 

41  0995  10.0483 

1172  8.5340 

1349  7.4124 

1527  6.5478 

1706  5.8605  19 

42  0998  10.0187 

1175  8.5126 

1352  7.3962 

1530  6.5350 

1709  5.8502  18 

43  i  1001  9.9893 

1178  8.4913 

1355  7-3800 

1533  6.5223 

1712  5.8400  17 

44  !  1004  9.9601 

1181  8.4701 

1358  7.3639 

1536  6.5097 

1715  5.8298  16 

45  '  1007  9.9310 

1184  8.4490 

1361  7.3479 

1539  6.4971 

1718  5.8197  15 

46  1010  9-9021 

1187  8.4280 

1864  7.3319 

1542  6.4846 

1721  5.8095  14 

47  1013  9.8734 

1189  8.4071 

1367  7.3160 

1545  6.4721 

1724  5.7994  13 

48  1016  9.8448 

1192  8.3863 

1370  7.3002 

1548  6.4596 

1727  5.7894  12 

49  1019  9.8164 

1195  8.3656 

1373  7.2844 

1551  6.4472 

1730  5.7794  11 

5O  1022  9.7882 

1198  8.3450 

1376  7.2687 

1554  6.4348 

1733  5.7694  1O 

51  1025  9.7601 

1201  8.3245 

1379  7.2531 

1557  6.4225 

1736  5.7594  9 

52  1028  9.7322 

1204  8.3041 

1382  7.2375 

1560  6.4103 

1739  5.7495  8 

53  |  1030  9.7044 

1207  8.2838 

1385  7.2220 

1563  6.3980 

1742  5.7396  7 

54 

1033  9.6768 

1210  8.2636 

1388  7.2066 

1566  6.3S.S9 

1745  5.7297  6 

55 

1036  9.6499 

1213  8.2434 

1391  7.1912 

1569  6.3737 

1748  5.7199  5 

56 

1039  9.6220 

1216  8.2234 

1394  7.1759 

1572  6.3617 

1751  5.7101  4 

57 

1042  9.5949 

1219  8.2035 

1397  7.1607 

1575  6.3496 

1754  5.7004  3 

58  1045  9.5679 

1222  8.1837 

1399  7.1455 

1578  6.3376 

1757  5.6906  2 

59  1048  9.5411 

1225  8.1640 

1402  7.1304 

1581  6.3257 

1760  5.6809   1 

6O  |  1051  9.5144 

1228  8.1443 

1405  7.1154 

1584  6.3138 

1763  5.6713  O 

cot    tan 

cot   tan 

cot    tan 

cot   tan 

cot    tan 

'     84° 

83° 

82 

81° 

80°     ' 

NATURAL   TANGENTS   AND   COTANGENTS. 


73 


11° 

12° 

13° 

14° 

f 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

0 

1763  5.6713 

1944  5.1446 

2126  4.7046 

2309  4.3315 

2493  4.0108 

<IO 

1 

1766  5.6617 

1947  5.1366 

2129  4.6979 

2312  4.3257 

24%  4.0058 

59 

2 

1769  5.6521 

1950  5.1286 

2132  4.6912 

2315  4.3200 

2499  4.0009 

58 

3 

1772  5.6425 

1953  5.1207 

2135  4.6845 

2318  4.3143 

2503  3.9959 

57 

4 

1775  5.6330 

1956  5.1128 

2138  4.6779 

2321  4.3086 

2506  3.9910 

56 

5 

1778  5.6234 

1959  5.1049 

2141  4.6712 

2324  4.3029 

2509  3.9861 

55 

6 

1781  5.6140 

1962  5.0970 

2144  4.6646 

2327  4.2972 

2512  3.9812 

54 

7 

1784  5.6045 

1965  5.0892 

2147  4.6580 

2330  4.2916 

2515  3.9763 

53 

8 

1787  5.5951 

1968  5.0814 

2150  4.6514 

2333  4.2859 

2518  3.9714 

52 

9 

1790  5.5857 

1971  5.0736 

2153  4.6448 

2336  4.2803 

252.1  3.9665 

51 

1O 

1793  5.5764 

1974  5.0658 

2156  4.6382 

2339  4.2747 

2524  3.9617 

5O 

11 

1796  5.5671 

1977  5.0581 

2159  4.6317 

2342  4.2691 

2527  3.9568 

49 

12 

1799  5.5578 

1980  5.0504 

2162  4.6252 

2345  4.2635 

2530  3.9520 

48 

13 

1802  5.5485 

1983  5.0427 

2165  4.6187 

2349  4.2580 

2533  3.9471 

47 

14 

1805  5.5393 

1986  5.0350 

2168  4.6122 

2352.  4.2524 

2537  3.9423 

46 

15 

1808  5.5301 

1989  5.0273 

2171  4.6057 

2355  4.2468 

2540  3.9375 

45 

16 

1811  5.5209 

1992  5.0197 

2174  4.5993 

2358  4.2413 

2543  3.9327 

44 

17 

1814  5.5118 

1995  5.0121 

2177  4.5928 

2361  4.2358 

2546  3.9279 

43 

18 

1817  5.5026 

1998  5.0045 

2180  4.5864 

2364  4.2303 

2549  3.9232 

42 

19 

1820  5.4936 

2001  4.9969 

2183  4.5800 

2367  4.2248 

2552  3.9184 

41 

2O 

1823  5.4845 

2004  4.9894 

2186  4.5736 

2370  4.2193 

2555  3.9136 

4O 

21 

1826  5.4755 

2007  4.9819 

2189  4.5673 

2373  4.2139 

2558  3.9089 

39 

22 

1829  5.4665 

2010  4.9744 

2193  4.5609 

2376  4.2084 

2561  3.9042 

38 

23 

1832  5.4575 

2013  4.9669 

21%  4.5546 

2379  4.2030 

2564  3.8995 

37 

24 

1835  5.4486 

2016  4.9594 

2199  4.5483 

2382  4.1976 

2568  3.8947 

36 

25 

1838  5.4397 

2019  4.9520 

2202  4.5420 

2385  4.1922 

2571  3.8900 

35 

26 

1841  5.4308 

2022  4.9446 

2205  4.5357 

2388  4.1868 

2574  3.8854 

34 

27 

1844  5.4219 

2025  4.9372 

2208  4.5294 

2392  4.1814 

2577  3.8807 

33 

28 

1847  5.4131 

2028  4.9298 

2211  4.5232 

2395  4.1760 

2580  3.8760 

32 

29 

1850  5.4043 

2031  4.9225 

2214  4.5169 

2398  4.1706 

2583  3.8714 

31 

30 

1853  5.3955 

2035  4.9152 

2217  4.5107 

2401  4.1653 

2586  3.8667 

3O 

31 

1856  5.3868 

2038  4.9078 

2220  4.5045 

2404  4.1600 

2589  3.8621 

29 

32 

1859  5.3781 

2041  4.9006 

2223  4.4983 

2407  4.1547 

2592  3.8575 

28 

33 

1862  5.3694 

2044  4.8933 

2226  4.4922 

2410  4.1493 

2595  3.8528 

27 

34 

1865  5.3607 

2047  4.8860 

2229  4.4860 

2413  4.1441 

2599  3.8482 

26 

35 

1868  5.3521 

2050  4.8788 

2232  4.4799 

2416  4.1388 

2602  3.8436 

25 

36 

1871  5.3435 

2053  4.8716 

2235  4.4737 

2419  4.1335 

2605  3.8391 

24 

37 

1874  5.3349 

2056  4.8644 

2238  4.4676 

2422  4.1282 

2608  3.8345 

23 

38 

1877  5.3263 

2059  4.8573 

2241  4.4615 

2425  4.1230 

2611  3.8299 

22 

39 

1880  5.3178 

2062  4.8501 

2244  4.4555 

2428  4.1178 

2614  3.8254 

21 

40 

1883  5.3093 

2065  4.8430 

2247  4.4494 

2432  4.1126 

2617  3.8208 

20 

41 

1887  5.3008 

2068  4.8359 

2251  4.4434 

2435  4.1074 

2620  3.8163 

19 

42 

1890  5.2924 

2071  4.8288 

2254  4.4374 

2438  4.1022 

2623  3.8118 

18 

43 

1893  5.2839 

2074  4.8218 

2257  4.4313 

2441  4.0970 

2627  3.8073 

17 

44 

1896  5.2755 

2077  4.8147 

2260  4.4253 

2444  4.0918 

2630  3.8028 

16 

45 

1899  5.2672 

2080  4.8077 

2263  4.4194 

2447  4.0867 

2633  3.7983 

15 

46 

1902  5.2588 

2083  4.8007 

2266  4.4134 

2450  4.0815 

2636  3.7938 

14 

47 

1905  5.2505 

2086  4.7937 

2269  4.4075 

2453  4.0764 

2639  3.7893 

13 

48 

1908  5.2422 

2089  4.7867 

2272  4.4015 

2456  4.0713 

2642  3.7848 

12 

49 

1911  5.2339 

2092  4.7798 

2275  4.3956 

2459  4.0662 

2645  3.7804 

11 

5O 

1914  5.2257 

2095  4.7729 

2278  4.3897 

2462  4.0611 

2648  3.7760 

10 

51 

1917  5.2174 

2098  4.7659 

2281  4.3838 

2465  4.0560 

2651  3.7715 

9 

52 

1920  5.2092 

2101  4.7591 

2284  4.3779 

2469  4.0509 

2655  3.7671 

8 

53 

1923  5.2011 

2104  4.7522 

2287  4.3721 

2472  4.0459 

2658  3.7627 

7 

54 

1926  5.1929 

2107  4.7453 

2290  4.3662 

2475  4.0408 

2661  3.7583 

6 

55 

1929  5.1848 

2110  4.7385 

2293  4.3604 

2478  4.0358 

2664  3.7539 

5 

56 

1932  5.1767 

2113  4.7317 

22%  4.3546 

2481  4.0308 

2667  3.7495 

4 

57 

1935  5.1686 

2116  4.7249 

2299  4.3488 

2484  4.0257 

2670  3.7451 

3 

58 

1938  5.1606 

2119  4.7181 

2303  4.3430 

2487  4.0207 

2673  3.7408 

2 

59 

1941  5.1526 

2123  4.7114 

2306  4.3372 

2490  4.0158 

2676  3.7364 

1 

6O 

1944  5.1446 

2126  4.7046 

2309  4.3315 

2493  4.0108 

2679  3.7321 

0 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

/ 

79° 

78° 

77° 

7<i 

75° 

f 

74 


NATURAL   TANGENTS   AND   COTANGENTS. 


t 

15° 

16° 

17° 

18° 

19° 

t 

tan   cot 

tan    cot 

tan   cot 

tan   cot 

tan    cot 

o 

2679  3.7321 

2867  3.4874 

3057  3.2709 

3249  3.0777 

3443  2.9042 

6O 

1 

2683  3.7277 

2871  3.4836 

3060  3.2675 

3252  3.0746 

3447  2.9015 

59 

2 

2686  3.7234 

2874  3.4798 

3064  3.2641 

3256  3.0716 

3450  2.8987 

58 

3 

2689  3.7191 

2877  3.4760 

3067  3.2607 

3259  3.0686 

3453  2.8960 

57 

4 

2692  3.7148 

2880  3.4722 

3070  3.2573 

3262  3.0655 

3456  2.8933 

56 

5 

2695  3.7105 

2883  3.4684 

3073  3.2539 

3265  3.0625 

3460  2.8905 

55 

6 

2698  3.7062 

2886  3.4646 

3076  3.2506 

3269  3.0595 

3463  2.8878 

54 

7 

2701  3.7019 

2S90  3.4608 

3080  3.2472 

3272  3.0565 

3466  2.8851 

53 

8 

2704  3.6976 

2893  3.4570 

3083  3.2438 

3275  3.0535 

3469  2.8824 

52 

9 

2708  3.6933 

2896  3.4533 

3086  3.2405 

3278  3.0505 

3473  2.8797 

51 

1O 

2711  3.6891 

2899  3.4495 

3089  3.2371 

3281  3.0475 

3476  2.8770 

5O 

11 

2714  3.6848 

2902  3.4458 

3092  3.2338 

3285  3.0445 

3479  2.8743 

49 

12 

2717  3.6806 

2905  3.4420 

3096  3.2305 

3288  3.0415 

3482  2.8716 

48 

13 

2720  3.6764 

2908  3.4383 

3099  3.2272 

3291  3.0385 

3486  2.8689 

47 

14 

2723  3.6722 

2912  3:4346 

3102  3.2238 

3294  3.0356 

3489  2.8662 

46 

15 

2726  3.6680 

2915  3.4308 

3105  3.2205 

3298  3.0326 

3492  2.8636 

45 

16 

2729  3.6638 

2918  3.4271 

3108  3.2172 

3301  3.0296 

3495  2.8609 

44 

17 

2733  3.6596 

2921  3.4234 

3111  3.2139 

3304  3.0267 

3499  2.8582 

43 

18 

2736  3.6554 

2924  3.4197 

3115  3.2106 

3307  3.0237 

3502  2.8556 

42 

19 

2739  3.6512 

2927  3.4160 

3118  3.2073 

3310  3.0208 

3505  2.8529 

41 

2O 

2742  3.6470 

2931  3.4124 

3121  3.2041 

3314  3.0178 

3508  2.8502 

4O 

21 

2745  3.6429 

2934  3.4087 

3124  3.2008 

3317  3.0149 

3512  2.8476 

39 

22 

2748  3.6387 

2937  3.4050 

3127  3.1975 

3320  3.0120 

3515  2.8449 

38 

23 

2751  3.6346 

2940  3.4014 

3131  3.1943 

3323  3.0090 

3518  2.8423 

37 

24 

2754  3.6305 

2943  3.3977 

3134  3.1910 

3327  3.0061 

3522  2.8397 

36 

25 

2758  3.6264 

2946  3.3941 

3137  3.1878 

3330  3.0032 

3525  2.8370 

35 

26 

2761  3.6222 

2949  3.3904 

3140  3.1845 

3333  3.0003 

3528  2.8344 

34 

27 

2764  3.6181 

2953  3.3868 

3143  3.1813 

3336  2.9974 

3531  2.8318 

33 

28 

2767  3.6140 

2956  3.3832 

3147  3.1780 

3339  2.9945 

3535  2.8291 

32 

29 

2770  3.6100 

2959  3.3796 

3150  3.1748 

3343  2.9916 

3538  2.8265 

31 

30 

2773  3.6059 

2962  3.3759 

3153  3.1716 

3346  2.9887 

3541  2.8239 

3O 

31 

2776  3.6018 

2965  3.3723 

3156  3.1684 

3349  2.9858 

3544  2.8213 

29 

32 

2780  3.5978 

2968  3.3687 

3159  3.1652 

3352  2.9829 

3548  2.S187 

28 

33 

2783  3.5937 

2972  3.3652 

3163  3.1620 

3356  2.9800 

3551  2.8161 

27 

34 

2786  3.5897 

2975  3.3616 

3166  3.1588 

3359  2.9772 

3554  2.8135  26 

35 

2789  3.5856 

2978  3.3580 

3169  3.1556 

3362  2.9743 

3558  2.8109 

25 

36 

2792  3.5816 

2981  3.3544 

3172  3.1524 

3365  2.9714 

3561  2.8083 

24 

37 

2795  3.5776 

2984  3.3509 

3175  3.1492 

3369  2.9686 

3564  2.8057 

23 

38 

2798  3.5736 

2987  3.3473 

3179  3.1460 

3372  2.9657 

3567  2.S032 

22 

39 

2801  3.5696 

2991  3.3438 

3182  3.1429 

3375  2.9629 

3571  2.8006 

21 

4O 

2805  3.5656 

2994  3.3402 

3185  3.1397 

3378  2.9600 

3574  2.7980 

2O 

41 

2808  3.5616 

2997  3.3367 

3188  3.1366 

3382  2.9572 

3577  2.7955 

19 

42 

2811  3.5576 

3000  3.3332 

3191  3.1334 

3385  2.9544 

3581  2.7929 

18 

43 

2814  3.5536 

3003  3.3297 

3195  3.1303 

3388  2.9515 

3584  2.7903 

17 

44 

2817  3.5497 

3006  3.3261 

3198  3.1271 

3391  2.9487 

3587  2.7878 

16 

45 

2820  3.5457 

3010  3.3226 

3201  3.1240 

3395  2.9459 

3590  2.7852 

15 

46 

2823  3.5418 

3013  3.3191 

3204  3.1209 

3398  2.9431 

3594  2.7827 

14 

47 

2827  3.5379 

3016  3.3156 

3207  3.1178 

3401  2.9403 

3597  2.7801 

13 

48 

2830  3.5339 

3019  3.3122 

3211  3.1146 

3404  2.9375 

3600  2.7776 

12 

49 

2833  3.5300 

3022  3.3087 

3214  3.1115 

3408  2.9347 

3604  2.7751 

11 

5O 

2836  3.5261 

3026  3.3052 

3217  3.1084 

3411  2.9319 

3607  2.7725 

10 

51 

2839  3.5222 

3029  3.3017 

3220  3.1053 

3414  2.9291 

3610  2.7700 

9 

52 

2842  3.5183 

3032  3.2983 

3223  3.1022 

3417  2.9263 

3613  2.7675 

8 

53 

2845  3.5144 

3035  3.2948 

3227  3.0991 

3421  2.9235 

3617  2.7650 

7 

54 

2849  3.5105 

3038  3.2914 

3230  3.0961 

3424  2.9208 

3620  2.7625 

6 

55 

2852  3.5067 

3041  3.2880 

3233  3.0930 

3427  2.9180 

3623  2.7500 

5 

56 

2855  3.5028 

3045  3.2845 

3236  3.0899 

3430  2.9152 

3627  2.7575 

4 

57 

2858  3.4989 

3048  3.2811 

3240  3.0868 

3434  2.9125 

3630  2.7550 

3 

58 

2861  3.4951 

3051  3.2777 

3243  3.0838 

3437  2.9097 

3633  2.7525 

2 

59 

2864  3.4912 

3054  3.2743 

3246  3.0S07- 

3440  2.9070 

3636  2.7500 

1 

60 

2867  3.4874 

3057  3.2709 

3249  3.0777 

3443  2.9042 

3640  2.7475   O 

cot   tan 

cot   tan 

cot   tan 

cot    tan 

cot   tan 

/     74° 

73° 

72° 

71° 

70°     ' 

NATURAL  TANGENTS  AND  COTANGENTS. 


75 


'     20° 

21° 

22° 

23° 

24°    1  ' 

tan   cot 

tail   cot 

tan   cot 

tan   cot 

tan   cot 

o 

3640  2.7475 

3839  2.6051 

4040  2.4751 

4245  2.3559 

4452  2.2460 

6O 

1 

3643  2.7450 

3842  2.6028 

4044  2.4730 

4248  2.3539 

4456  2.2443 

59 

2 

3646  2.7425 

3845  2.6006 

4047  2.4709 

4252  2.3520 

4459  2.2425 

58 

3 

3650  2.7400. 

3849  2.5983 

4050  2.4689 

4255  2.3501 

4463  2.2408 

57 

4 

3653  2.7376 

3852  2.5961 

4054  2.4668 

4258  2.3483 

4466  2.2390 

56 

5 

3656  2.7351 

3855  2.5938 

4057  2.4648 

4262  2.3464 

4470  2.2373 

55 

6 

3659  2.7326 

3859  2.5916 

4061  2.4627 

4265  2.3445 

4473  2.2355 

54 

7 

3663  2.7302 

3862  2.5893 

4064  2.4606 

4269  2.3426 

4477  2.2338 

53 

8 

3666  2.7277 

3865  2.5871 

4067  2.4586 

4272  2.3407 

4480  2.2320 

52 

9 

3669  2.7253 

3869  2.5848 

4071  2.4566 

4276  2.3388 

4484  2.2303 

51 

1O 

3673  2.7228 

3872  2.5826 

4074  2.4545 

4279  2.3369 

4487  2.2286 

5O 

11 

3676  2.7204 

3875  2.5804 

4078  2.4525 

4283  2.3351 

4491  2.2268 

49 

12 

3679  2.7179 

3879  2.5782 

4081  2.4504 

4286  2.3332 

4494  2.2251 

48 

13 

3683  2.7155 

3882  2.5759 

4084  2.4484 

4289  2.3313 

4498  2.2234 

47 

14 

3686  2.7130 

3885  2.5737 

4088  2.4464 

4293  2.3294 

4501  2.2216 

46 

15 

3689  2.7106 

3889  2.5715 

4091  2.4443 

4296  2.3276 

4505  2.2199 

45 

16 

3693  2.7082 

3892  2.5693 

4095  2.4423 

4300  2.3257 

4508  2.2182 

44 

17 

3696  2.7058 

3895  2.5671 

4098  2.4403 

4303  2.3238 

4512  2.2165 

43 

18 

3699  2.7034 

3899  2.5649 

4101  2.4383 

4307  2.3220 

4515  2.2148 

42 

19 

3702  2.7009 

3902  2.5627 

4105  2.4362 

4310  2.3201 

4519  2.2130 

41 

2O 

3706  2.6985 

3906  2.5605 

4108  2.4342 

4314  2.3183 

4522  2.2113 

4O 

21 

3709  2.6961 

3909  2.5583 

4111  2.4322 

4317  2.3164 

4526  2.2096 

39 

22 

3712  2.6937 

3912  2.5561 

4115  2.4302 

4320  2.3146 

4529  2.2079 

38 

23 

3716  2.6913 

3916  2.5539 

4118  2.4282 

4324  2.3127 

4533  2.2062 

37 

24 

3719  2.6S89 

3919  2.5517 

4122  2.4262 

4327  2.3109 

4536  2.2045 

36 

25 

3722  2.6865 

3922  2.5495 

4125  2.4242 

4331  2.3090 

4540  2.2028 

35 

26 

3726  2.6841 

3926  2.5473 

4129  2.4222 

4334  2.3072 

4543  2.2011 

34 

27 

3729  2.6818 

3929  2.5452 

4132  2.4202 

4338  2.3053 

4547  2.1994 

33 

28 

3732  2.6794 

3932  2.5430 

4135  2.4182 

4341  2.3035 

4550  2.1977 

32 

29 

3736  2.6770 

3936  2.5408 

4139  2.4162 

4345  2.3017 

4554  2.1960 

31 

30 

3739  2.6746 

3939  2.53S6 

4142  2.4142 

4348  2.2998 

4557  2.1943 

3O 

31 

3742  2.6723 

3942  2.5365 

4146  2.4122 

4352  2.2980 

4561  2.1926 

29 

32 

3745  2.6699 

3946  2.5343 

4149  2.4102 

4355  2.2962 

4564  2.1909 

28 

33 

3749  2.6675 

3949  2.5322 

4152  2.4083 

4359  2.2944 

4568  2.1892 

27 

34 

3752  2.6652 

3953  2.5300 

4156  2.4063 

4362  2.2925 

4571  2.1876 

26 

35 

3755  2.6628 

3956  2.5279 

4159  2.4043 

4365  2.2907 

4575  2.1859 

25 

36  3759  2.6605 

3959  2.5257 

4163  2.4023 

4369  2.2889 

4578  2.1842 

24 

37  3762  2.6581 

3963  2.5236 

4166  2.4004 

4372  2.2871 

4582  2.1825 

23 

38  !  3765  2.65:8 

3966  2.5214 

4169  2.3984 

4376  2.2853 

4585  2.1808 

22 

39  3769  2.6534 

3969  2.5193 

4173  2.3964 

4379  2.2835 

4589  2.1792 

21 

4O 

3772  2.6511 

3973  2.5172 

4176  2.3945 

4383  2.2817 

4592  2.1775 

2O 

41 

3775  2.648S 

3976  2.5150 

4180  2.3925 

4386  2.2799 

45%  2.1758 

19 

42 

3779  2.6464 

3979  2.5129 

4183  2.3906 

4390  2.2781 

4599  2.1742 

18 

43 

3782  2.6441 

3983  2.5108 

4187  2.3886 

4393  2.2763 

4603  2.1725 

17 

44 

3785  2.6418 

3986  2.5086 

4190  2.3867 

4397  2.2745 

4607  2.1708 

16 

45 

3789  2.6395 

3990  2.5065 

4193  2.3847 

4400  2.2727 

4610  2.1692 

15 

46 

3792  2.6371 

3993  2.5044 

4197  2.3828 

4404  2.2709 

4614  2.1675 

14 

47 

3795  2.6348 

3996  2.5023 

4200  2.3808 

4407  2.2691 

4617  2.1659 

13 

48 

3799  2.6325 

4000  2.5002 

4204  2.3789 

4411  2.2673 

4621  2.1642 

12 

49  3802  2.6302 

4003  2.4981 

4207  2.3770 

4414  2.2655 

4624  2.1625 

11 

5O  3805  2.6279 

4006  2.4960 

4210  2.3750 

4417  2.2637 

4628  2.1609 

10 

51   3809  2.6256 

4010  2.4939 

4214  2.3731 

4421  2.2620 

4631  2.1592 

9 

52  3812  2.6233 

4013  2.4918 

4217  2.3712 

4424  2.2602 

4635  2.1576 

8 

53   3815  2.6210 

•1017  2.4897 

4221  2.3693 

4428  2.2584 

4638  2.1560 

7 

54 

3819  2.6187 

4020  2.4876 

4224  2.3673 

4431  2.2566 

4642  2.1543 

6 

55 

3822  2.6165 

4023  2.4855 

4228  2.3654 

4435  2.2549 

4645  2.1527 

5 

56 

3825  2.6142 

4027  2.4834 

4231  2.3635 

4438  2.2531 

4649  2.1510 

4 

57 

3829  2.6119 

4030  2.4813 

4234  2.3616 

4442  2.2513 

4652  2.1494 

3 

58 

3832  2.6096 

4033  2.4792 

4238  2.3597 

4445  2.24% 

4656  2.1478 

2 

59 

3835  2.6074 

4037  2.4772 

4241  2.3578 

4449  2.2478 

4660  2.1461 

1 

6O  3839  2.6051 

4040  2.4751 

4245  2.3559 

4452  2.2460 

4663  2.1445 

0 

1  cot    tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

'     69° 

68° 

67° 

66° 

«5C 

i 

76 


NATURAL  TANGENTS  AND  COTANGENTS. 


f 

25° 

26° 

27° 

28C 

29° 

t 

tan   cot 

tan   cot 

tan    cot 

tan   cot 

tan   cot 

o 

4663  2.1445 

4877  2.0503 

5095  1.9626 

5317  1.8807 

5543  1.8040 

60 

1 

4667  2.1429 

4881  2.0488 

5099  1.9612 

5321  1.8794 

5547  1.8028 

59 

2 

4670  2.1413 

4885  2.0473 

5103  1.9598 

5325  1.8781 

5551  1.8016 

58 

3 

4674  2.1396 

4888  2.0458 

5106  1.9584 

5328  1.8768 

5555  1.8003 

57 

4 

4677  2.1380 

4892  2.0443 

5110  1.9570 

5332  1.8755 

5558  1.7991 

56 

5 

4681  2.1364 

4895  2.0428 

5114  1.9556 

5336  1.8741 

5562  1.7979 

55 

6 

4684  2.1348 

4899  2.0413 

5117  1.9542 

5340  1.8728 

5566  1.7966 

54 

7 

4688  2.1332 

4903  2.0398 

5121  1.9528 

5343  1.8715 

5570  1.7954 

53 

8 

4691  2.1315 

4906  2.0383 

5125  1.9514 

5347  1.8702 

5574  1.7942 

52 

9 

4695  2.1299 

4910  2.0368 

5128  1.9500 

5351  1.8689 

5577  1.7930 

51 

1O 

4699  2.1283 

4913  2.0353 

5132  1.9486 

5354  1.8676 

5581  1.7917 

50 

11 

4702  2.1267 

4917  2.0338 

5136  1.9472 

5358  1.8663 

5585  1.7905 

49 

12 

4706  2.1251 

4921  2.0323 

5139  1.9458 

5362  1.8650 

5589  1.7893 

48 

13 

4709  2.1235 

4924  2.0308 

5143  1.9444 

5366  1.8637 

5593  1.7881 

47 

14 

4713  2.1219 

4928  2.0293 

5147  1.9430 

5369  1.8624 

5596  1.7868 

46 

15 

4716  2.1203 

4931  2.0278 

5150  1.9416 

5373  1.8611 

5600  1.7856 

45 

16 

4720  2.1187 

4935  2.0263 

5154  1.9402 

5377  1.8598 

5604  1.7844 

44 

17 

4723  2.1171 

4939  2.0248 

5158  1.9388 

5381  1.8585 

5608  1.7832 

43 

18 

4727  2.1155 

4942  2.0233 

5161  1.9375 

5384  1.8572 

5612  1.7820 

42 

19 

4731  2.1139 

4946  2.0219 

5165  1.9361 

5388  1.8559 

5616  1.7808 

41 

2O 

4734  2.1123 

4950  2.0204 

5169  1.9347 

5392  1.8546 

5619  1.7796 

4O 

21 

4738  2.1107 

4953  2.0189 

5172  1.9333 

5396  1.8533 

5623  1.7783 

39 

22 

4741  2.1092 

4957  2.0174 

5176  1.9319 

5399  1.8520 

5627  1.7771 

38 

23 

4745  2.1076 

4960  2.0160 

5180  1.9306 

5403  1.8507 

5631  1.7759 

37 

24 

4748  2.1060 

4964  2.0145 

5184  1.9292 

5407  1.8495 

5635  1.7747 

36 

25 

4752  2.1044 

4968  2.0130 

5187  1.9278 

5411  1.8482 

5639  1.7735 

35 

26 

4755  2.1028 

4971  2.0115 

5191  1.9265 

5415  1.8469 

5642  1.7723  :  34 

27 

4759  2.1013 

4975  2.0101 

5195  1.9251 

5418  1.8456 

5646  1.7711  !  33 

28 

4763  2.0997 

4979  2.0086 

5198  1.9237 

5422  1.8443 

5650  1.7699 

32 

29 

4766  2.0981 

4982  2.0072 

5202  1.9223 

5426  1.8430 

5654  1.7687 

31 

30 

4770  2.0965 

4986  2.0057 

5206  1.9210 

5430-1.8418 

5658  1.7675 

3O 

31 

4773  2.0950 

4989  2.0042 

5209  1.9196 

5433  1.8405 

5662  1.7663 

29 

32 

4777  2.0934 

4993  2.0028 

5213  1.9183 

5437  1.8392 

5665  1.7651 

28 

33 

4780  2.0918 

4997  2.0013 

5217  1.9169 

5441  1.8379 

5669  1.7639 

27 

34 

4784  2.0903 

5000  1.9999 

5220  1.9155 

5445  1.8367 

5673  1.7627 

26 

35 

4788  2.0887 

5004  1.9984 

5224  1.9142 

5448  1.8354 

5677  1.7615 

25 

36 

4791  2.0872 

5008  1.9970 

5228  1.9128 

5452  1.8341 

5681  1.7603 

24 

37 

4795  2.0856 

5011  1.9955 

5232  1.9115 

5456  1.8329 

5685  1.7591 

23 

38 

4798  2.0840 

5015  1.9941 

5235  1.9101 

5460  1.8316 

5688  1.7579  !  22 

39 

4802  2.0825 

5019  1.9926 

5239  1.9088 

5464  1.8303 

5692  1.7567  21 

40 

4806  2.0809 

5022  1.9912 

5243  1.9074 

5467  1.8291 

5696  1.7556  2O 

41 

4S09  2.0794 

5026  1.9897 

5246  1.9061 

5471  1.8278 

5700  1.7544   19 

42 

4813  2.0778 

5029  1.9883 

5250  1.9047 

5475  1.8265 

5704  1.7532  18 

43 

4816  2.0763 

5033  1.9868 

5254  1.9034 

5479  1.8253 

5708  .7520  |  17 

44 

4820  2.0748 

5037  1.9854 

5258  1.9020 

5482  1.8240 

5712  1.7508 

16 

45 

4823  2.0732 

5040  1.9840 

5261  1.9007 

5486  1.8228 

5715  .7496 

15 

46 

4827  2.0717 

5044  1.9825 

5265  1.8993 

5490  1.8215 

5719  .7485 

14 

47 

4831  2.0701 

5048  1.9811 

5269  1.8980 

5494  1.8202 

5723  .7473 

13 

48 

4834  2.0686 

5051  1.9797 

5272  1.8967 

5498  1.8190 

5727  .7461 

12 

49 

4838  2.0671 

5055  1.9782 

5276  1.8953 

5501  1.8177 

5731  .7449 

11 

5O 

4841  2.0655 

5059  1.9768 

5280  1.8940 

5505  1.8165 

5735  .7437  1O 

51 

4845  2.0640 

5062  1.9754 

5284  1.8927 

5509  1.8152 

5739  .7426   9 

52 

4849  2.0625 

5066  1.9740 

5287  1.8913 

5513  1.8140 

5743  .7414   8 

53 

4852  2.0609 

5070  1.9725 

5291  1.8900 

5517  1.8127 

5746  .7402 

7 

54 

4856  2.0594 

5073  1.9711 

5295  1.8887 

5520  1.8115 

5750  .7391 

6 

55 

4859  2.0579 

5077  1.9697 

5298  1.8873 

5524  1.8103 

5754  .7379 

5 

56 

4863  2.0564 

5081  1.9683 

5302  1.8860 

5528  1.8090 

5758  .7367 

4 

57 

4867  2.0549 

5084  1.9669 

5306  1.8847 

5532  1.8078 

5762  .7355 

3 

58 

4870  2.0533 

5088  1.9654 

5310  1.8834 

5535  1.8065 

5766  .7344 

2 

59 

4874  2.0518 

5092  1.9640 

5313  1.SS20 

5539  1.8053 

5770  .7332 

1 

60 

4877  2.0503 

5095  1.9626 

5317  1.8807 

5543  1.8040 

5774  1.7321 

0 

cot    tan 

cot   tan 

cot    tan 

cot   tan 

cot   tan 

f 

64° 

68° 

62° 

61° 

60°     ' 

NATURAL  TANGENTS  AND  COTANGENTS. 


77 


' 

3O° 

31° 

32° 

33C 

34° 

r 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

0 

5774  1.7321 

6009  1.6643 

6249  1.6003 

6494  1.5399 

6745  1.4826  !«O 

1 

5777  1.7309 

6013  1.6632 

6253  1.5993 

6498  1.5389 

6749  1.4816 

59 

2 

5781  1.7297 

6017  1.6621 

6257  1.5983 

6502  1.5379 

6754  1.4807 

58 

3 

5785  1.7286 

6020  1.6610 

6261  1.5972 

6506  1.5369 

6758  1.4798 

57 

4 

5789  1.7274 

6024  1.6599 

6265  1.5962 

6511  1.5359 

6762  1.4788 

56 

5 

5793  1.7262 

6028  1.6588 

6269  1.5952 

6515  1.5350 

6766  1.4779 

55 

6 

5797  1.7251 

6032  1.6577 

6273  1.5941 

6519  1.5340 

6771  1.4770 

54 

7 

5801  1.7239 

6036  1.6566 

6277  1.5931 

6523  1.5330 

6775  1.4761 

53 

8 

5805  .7228 

6040  1.6555 

6281  1.5921 

6527  1.5320 

6779  1.4751 

52 

9 

5808  1.7216 

6044  1.6545 

6285  1.5911 

6531  1.5311 

6783  1.4742 

51 

1O 

5812  .7205 

6048  1.6534 

6289  1.5900 

6536  1.5301 

6787  1.4733 

5O 

11 

5816  1.7193 

6052  1.6523 

6293  1.5890 

6540  1.5291 

6792  1.4724 

49 

12 

5820  .7182 

6056  1.6512 

6297  1.5880 

6544  1.5282 

67%  1.4715 

48 

13 

5824  .7170 

6060  1.6501 

6301  1.5869 

6548  1.5272 

6800  1.4705 

47 

14 

5828  1.7159 

6064  1.6490 

6305  1.5859 

6552  1.5262 

6805  1.46% 

46 

15 

5832  1.7147 

6068  1.6479 

6310  1.5849 

6556  1.5253 

6809  1.4687 

45 

16 

5836  1.7136 

6072  1.6469 

6314  1.5839 

6560  1.5243 

6813  1.4678 

44 

17 

5840  1.7124 

6076  1.6458 

6318  1.5829 

6565  1.5233 

6817  1.4669 

43 

18 

5844  1.7113 

6080  1.6447 

6322  1.5818 

6569  1.5224 

6822  1.4659 

42 

19 

5847  1.7102 

6084  1.6436 

6326  1.5808 

6573  1.5214 

6826  1.4650 

41 

20 

5851  1.7090 

6088  1.6426 

6330  1.5798 

6577  1.5204 

6830  1.4641 

40 

21 

5855  1.7079 

6092  1.6415 

6334  1.5788 

6581  1.5195 

6834  1.4632 

39 

22 

5859  1.7067 

6096  1.6404 

6338  1.5778 

6585  1.5185 

6839  1.4623 

38 

23 

5863  1.7056 

6100  1.6393 

6342  1.5768 

6590  1.5175 

6843  1.4614 

37 

24 

5867  1.7045 

6104  1.6383 

6346  1.5757 

6594  1.5166 

6847  1.4605 

36 

25 

5871  1.7033 

6108  1.6372 

6350  1.5747 

6598  1.5156 

6851  1.45% 

35 

26 

5875  1.7022 

6112  1.6361 

6354  1.5737 

6602  1.5147 

6856  1.4586 

34 

27 

5879  1.7011 

6116  1.6351 

6358  1.5727 

6606  1.5137 

6860  1.4577 

33 

28 

5883  1.6999 

6120  1.6340 

6363  1.5717 

6610  1.5127 

6864  1.4568 

32 

29 

5887  1.6988 

6124  1.6329 

6367  1.5707 

6615  1.5118 

6869  1.4559 

31 

30 

5890  1.6977 

6128  1.6319 

6371  1.5697 

6619  1.5108 

6873  1.4550 

30 

31 

5894  1.6965 

6132  1.6308 

6375  1.5687 

6623  1.5099 

6877  1.4541 

29 

32 

5898  1.6954 

6136  1.6297 

6379  1.5677 

6627  1.5089 

6881  1.4532 

28 

33 

5902  1.6943 

6140  1.6287 

6383  1.5667 

6631  1.5080 

6886  1.4523 

27 

34 

5906  1.6932 

6144  1.6276 

6387  1.5657 

6636  1.5070 

6890  1.4514 

26 

35 

5910  1.6920 

6148  1.6265 

6391  1.5647 

6640  1.5061 

6894  1.4505 

25 

36 

5914  1.6909 

6152  1.6255 

6395  1.5637 

6644  1.5051 

6899  1.44% 

24 

37 

5918  1.6898 

6156  1.6244 

6399  1.5627 

6648  1.5042 

6903  1.4487 

23 

38 

5922  1.6887 

6160  1.6234 

6403  1.5617 

6652  1.5032 

6907  1.4478 

22 

39 

5926  1.6875 

6164  1.6223 

6408  1.5607 

6657  1.5023 

6911  1.4469 

21 

40 

5930  1.6864 

6168  1.6212 

6412  1.5597 

6661  1.5013 

6916  1.4460 

2O 

41 

5934  1.6853 

6172  1.6202 

6416  1.5587 

6665  1.5004 

6920  1.4451 

19 

42 

5938  1.6842 

6176  1.6191 

6420  1.5577 

6669  1.4994 

6924  1.4442 

18 

43 

5942  1.6831 

6180  1.6181 

6424  1.5567 

6673  1.4985 

6929  1.4433 

17 

44 

5945  1.6820 

6184  1.6170 

6428  1.5557 

6678  1.4975 

6933  1.4424 

16 

45 

5949  1.6808 

6188  1.6160 

6432  1.5547 

6682  1.4966 

6937  1.4415 

15 

46 

5953  1.6797 

6192  1.6149 

6436  1.5537 

6686  1.4957 

6942  1.4406 

14 

47 

5957  1.6786 

61%  1.6139 

6440  1.5527 

6690  1.4947 

6946  1.4397 

13 

48 

5961  1.6775 

6200  1.6128 

6445  1.5517 

6694  1.4938 

6950  1.4388 

12 

49 

5965  1.6764 

6204  1.6118 

6449  1.5507 

6699  1.4928 

6954  1.4379 

11 

50 

5969  1.6753 

6208  1.6107 

6453  1.5497 

6703  1.4919 

6959  1.4370 

1O 

51 

5973  1.6742 

6212  1.6097 

6457  1.5487 

6707  1.4910 

6963  1.4361 

9 

52 

5977  1.6731 

6216  1.6087 

6461  1.5477 

6711  1.4900 

6%7  1.4352 

8 

53 

5981  1.6720 

6220  1.6076 

6465  1.5468 

6716  1.4891 

6972  1.4344 

7 

54 

5985  1.6709 

6224  1.6066 

6469  1.5458 

6720  1.4882 

6976  1.4335 

6 

55 

5989  1.6698 

6228  1.6055 

6473  1.5448 

6724  1.4872 

6980  1.4326 

5 

56 

5993  1.6687 

6233  1.6045 

6478  1.5438 

6728  1.4863 

6985  1.4317 

4 

57 

5997  1.6676 

6237  1.6034 

6482  1.5428 

6732  1.4854 

6989  1.4308 

3 

58 

6001  1.6665 

6241  1.6024 

6486  1.5418 

6737  1.4844 

6993  1.4299 

2 

59 

6005  1.6654 

6245  1.6014 

6490  1.5408 

6741  1.4835 

6998  1.4290 

1 

6O 

6009  1.6643 

6249  1.6003 

6494  1.5399 

6745  1.4826 

7002  1.4281 

0 

cot   tan 

cot   tan 

cot    tan 

cot   tan 

cot   tan 

t 

59° 

58° 

57° 

50° 

55° 

t 

78 


NATURAL   TANGENTS   AND   COTANGENTS. 


/ 

35° 

36° 

37° 

38° 

39°     9 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan 

cot 

o 

7002  1.4281 

7265  1.3764 

7536  1.3270 

7813  1.2799 

8098 

1.2349  6O 

1 

7006  1.4273 

7270  1.3755 

7540  1.3262 

7818  1.2792 

8103 

1.2342   59 

2 

7011  1.4264 

7274  1.3747 

7545  1.3254 

7822  1.2784 

8107 

1.2334   58 

3 

7015  1.4255 

7279  1.3739 

7549  1.3246 

7827  1.2776 

8112 

1.2327   57 

4 

7019  1.4246 

7283  1.3730 

7554  1.3238 

7832  1.2769 

8117 

1.2320  56 

5 

7024  1.4237 

7288  1.3722 

7558  1.3230 

7836  1.2761 

8122 

1.2312   55 

6 

7028  1.4229 

7292  1.3713 

7563  1.3222 

7841  1.2753 

8127 

1.2305   54 

7 

7032  1.4220 

7297  1.3705 

7568  1.3214 

7846  1.2746 

8132 

1.2298  53 

8 

7037  1.4211 

7301  1.3697 

7572  1.3206 

7850  1.2738 

8136 

1.2290  52 

9 

7041  1.4202 

7306  1.3688 

7577  1.3198 

7855  1.2731 

8141 

1.2283   51 

1O 

7046  1.4193 

7310  1.36SO 

7581  1.3190 

7860  .2723 

8146 

1.2276  5O 

11 

7050  1.4185 

7314  1.3672 

7586  1.3182 

7865  .2715 

8151 

1.2268  49 

12 

7054  1.4176 

7319  1.3663 

7590  1.3175 

7869  .2708 

8156 

1.2261   48 

13 

7059  1.4167 

7323  1.3655 

7595  1.3167 

7874  .2700 

8161 

1.2254  47 

14 

7063  1.4158 

7328  1.3647 

7600  1.3159 

7879  .2693 

8165 

1.2247  46 

15 

7067  1.4150 

7332  1.3638 

7604  1.3151 

7883  1.2685 

8170 

1.2239  45 

16 

7072  1.4141 

7337  1.3630 

7609  1.3143 

7888  1.2677 

8175 

1.2232  44 

17 

7076  1.4132 

7341  1.3622 

7613  1.3135 

7893  1.2670 

8180 

1.2225   43 

18 

7080  1.4124 

7346  1.3613 

7618  1.3127 

7898  1.2662 

8185 

1.2218  42 

19 

7085  1.4115 

7350  1.3605 

7623  1.3119 

7902  1.2655 

8190 

1.2210  41 

2O 

7089  1.4106 

7355  1.3597 

7627  1.3111 

7907  1.2647 

8195 

1.2203  4O 

21 

7094  1.4097 

7359  1.3588 

7632  1.3103 

7912  1.2640 

8199 

1.2196  39 

22 

7098  1.4089 

7364  1.3580 

7636  1.3095 

7916  1.2632 

8204 

1.2189  38 

23 

7102  1.4080 

7368  1.3572 

7641  1.3087 

7921  1.2624 

8209 

1.2181   37 

24 

7107  1.4071 

7373  1.3564 

7646  1.3079 

7926  1.2617 

8214 

1.2174  36 

25 

7111  1.4063 

7377  1.3555 

7650  1.3072 

7931  1.2609 

8219 

1.2167  35 

26 

7115  1.4054 

7382  1.3547 

7655  1.3064 

7935  1.2602 

8224 

1.2160  34 

27 

7120  1.4045 

7386  1.3539 

7659  1.3056 

7940  1.2594 

8229 

1.2153  33 

28 

7124  1.4037 

7391  1.3531 

7664  1.3048 

7945  1.2587 

8234 

1.2145  •  32 

29 

7129  1.4028 

7395  1.3522 

7669  1.3040 

7950  1.2579 

8238 

1.2138  31 

3O 

7133  1.4019 

7400  1.3514 

7673  1.3032 

7954  .2572 

8243 

1.2131  3O 

31 

7137  1.4011 

7404  1.3506 

7678  1.3024 

7959  1.2564 

8248 

1.2124   29 

32 

7142  1.4002 

7409  1.3498 

7683  1.3017 

7964  1.2557 

8253 

1.2117  |  28 

33 

7146  1.3994 

7413  1.3490 

7687  1.3009 

7969  1.2549 

8258 

1.2109  27 

34 

7151  1.3985 

7418  1.3481 

7692  1.3001 

7973  1.2542 

8263 

1.2102  i  26 

35 

7155  1.3976 

7422  1.3473 

7696  1.2993 

7978  1.2534 

8268 

1.2095   25 

36 

7159  1.3968 

7427  1.3465 

7701  1.2985 

7983  1.2527 

8273 

1.2088  24 

37 

7164  1.3959 

7431  1.3457 

7706  1.2977 

7988  1.2519 

8278 

1.2081   23 

38 

7168  1.3951 

7436  1.3449 

7710  1.2970 

7992  1.2512 

8283 

1.2074   22 

39 

7173  1.3942 

7440  1.3440 

7715  1.2962 

7997  1.2504 

8287 

1.2066  |  21 

4O 

7177  1.3934 

7445  1.3432 

7720  1.2954 

8002  1.2497 

8292 

1.2059  |2O 

41 

7181  1.3925 

7449  1.3424 

7724  1.2946 

8007  1.2489 

8297 

1.2052   19 

42 

7186  1.3916 

7454  1.3416 

7729  1.2938 

8012  1.2482 

8302 

1.2045   18 

43 

7190  1.3908 

7458  1.3408 

7734  1.2931 

8016  1.2475 

8307 

1.2038   17 

44 

7195  1.3899 

7463  1.3400 

7738  1.2923 

8021  1.2467 

8312 

1.2031   16 

45 

7199  13891 

7467  1.3392 

7743  1.2915 

8026  1.2460 

8317 

1.2024   15 

46 

7203  1.3882 

7472  1.3384 

7747  1.2907 

8031  1.2452 

8322 

1.2017   14 

47 

7208  1.3874 

7476  1.3375 

7752  1.2900 

8035  1.2445 

8327 

1.2009   13 

48  1  7212  1.3865 

7481  1.3367 

7757  1.2892 

8040  1.2437 

8332 

1.2002   12 

49  7217  1.3857 

7485  1.3359 

7761  1.2884 

8045  1.2430 

8337 

1.1995   11 

5O  7221  1.3848 

7490  1.3351 

7766  1.2876 

8050  1.2423 

8342 

1.1988  1O 

51   7226  1.3840 

7495  1.3343 

7771  1.2869 

8055  1.2415 

8346 

1.1981   9 

52   7230  1.3831 

7499  1.3335 

7775  1.2861 

8059  1.2408 

8351 

1.1974  :  8 

53  7234  1.3823 

7504  1.3327 

7780  1.2853 

8064  1.2401 

8356 

1.1967   7 

54  7239  1.3814 

7508  1.3319 

7785  1.2846 

8069  1.2393 

8361 

1.1960   6 

55  !  7243  1.3806 

7513  1.3311 

7789  1.2838 

8074  1.2386 

8366 

1.1953   5 

56   7248  1.3798 

7517  1.3303 

7794  1.2830 

8079  1.2378 

8371 

1.1946   4 

57   7252  1.3789 

7522  1.3295 

7799  1.2822 

8083  1.2371 

8376 

1.1939   3 

58   7257  1.3781 

7526  1.3287 

7803  1.2815 

8088  1.2364 

8381 

1.1932   2 

59   7261  1.3772 

7531  1.3278 

7808  1.2807 

8093  1.2356 

8386 

1.1925  j  1 

GO  7265  1.3764 

cot    tan 

7536  1.3270 

7813  1.2799 

8098  1.2349 

8391 

1.1918   O 

'     54° 

53° 

52° 

51C 

5O°     ' 

NATURAL   TANGENTS   AND   COTANGENTS. 


79 


t     40° 

41° 

42° 

43° 

44° 

t 

tan    cot 

tan    cot 

tan   cot 

tan   cot 

tan    cot 

o 

8391  1.1918 

8693  1.1504 

9004  1.1106 

9325  1.0724 

9657  1.0355 

6O 

1 

8396  1.1910 

8698  1.1497 

9009  1.1100 

9331  1.0717 

9663  1.0349 

59 

2 

8401  1.1903 

8703  1.1490 

9015  1.1093 

9336  1.0711 

9668  1.0343 

58 

3 

8406  1.1896 

8708  1.1483 

9020  1.1087 

9341  1.0705 

9674  1.0337 

57 

4 

8411  1.1889 

8713  1.1477 

9025  1.1080 

9347  1.0699 

9679  1.0331 

56 

5 

8416  1.1882 

8718  1.1470 

9030  1.1074 

9352  1.0692 

9685  1.0325 

55 

6 

8421  1.1875 

8724  1.1463 

9036  1.1067 

9358  1.0686 

9691  1.0319 

5! 

7 

8426  1.1868 

8729  1.1456 

9041  1.1061 

9363  1.0680 

9696  1.0313 

5: 

8 

8431  1.1861 

8734  1.1450 

9046  1.1054 

9369  1.0674 

9702  1.0307 

52 

9 

8436  1.1854 

8739  1.1443 

9052  1.1048 

9374  1.0668 

9708  1.0301 

51 

1O 

8441  1.1847 

8744  1.1436 

9057  1.1041 

9380  1.0661 

9713  1.0295 

5O 

11 

8446  1.1840 

8749  1.1430 

9062  1.1035 

9385  1.0655 

9719  1.0289 

49 

12 

8451  1.1833 

8754  1.1423 

9067  1.1028 

9391  1.0649 

9725  1.0283 

48 

13 

8456  1.1826 

8759  1.1416 

9073  1.1022 

9396  1.0643 

9730  1.0277 

47 

14 

8461  1.1819 

8765  1.1410 

9078  1.1016 

9402  1.0637 

9736  1.0271 

46 

15 

8466  1.1812 

8770  1.1403 

9083  1.1009 

9407  1.0630 

9742  1.0265 

45 

16 

8471  1.1806 

8775  1.1396 

9089  1.1003 

9413  1.0624 

9747  1.0259 

44 

17 

8476  1.1799 

8780  1.1389 

9094  1.0996 

9418  1.0618 

9753  1.0253 

43 

18 

8481  1.1792 

8785  1.1383 

9099  1.0990 

9424  1.06]  2 

9759  1.0247 

42 

19  8486  1.1785 

8790  1.1376 

9105  1.0983 

9429  1.0606 

9764  1.0241 

41 

20 

8491  1.1778 

8796  1.1369 

9110  1.0977 

9435  1.0599 

9770  1.0235 

4O 

21 

8496  1.1771 

8801  1.1363 

9115  1.0971 

9440  1.0593 

9776  1.0230 

39 

22 

8501  1.1764 

8806  1.1356 

9121  1.0964 

9446  1.0587 

9781  1.0224 

38 

23 

8506  1.1757 

8811  1.1349 

9126  1.0958 

9451  1.0581 

9787  1.0218 

37 

24 

8511  1.1750 

8816  1.1343 

9131  1.0951 

9457  1.0575 

9793  1.0212 

36 

25 

8516  1.1743 

8821  1.1336 

9137  1.0945 

9462  1.0569 

9798  1.0206 

35 

26 

8521  1.1736 

8827  1.1329 

9142  1.0939 

9468  1.0562 

9804  1.0200 

34 

27  8526  1.1729 

8832  1.1323 

9147  1.0932 

9473  1.0556 

9810  1.0194 

33 

28  8531  1.1722 

8837  1.1316 

9153  1.0926 

9479  1.0550 

9816  1.0188 

32 

29 

8536  1.1715 

8842  1.1310 

9158  1.0919 

9484  1.0544 

9821  1.0182 

31 

30 

8541  1.1708 

8847  1.1303 

9163  1.0913 

9490  1.0538 

9827  1.0176 

3O 

31 

8546  1.1702 

8852  1.1296 

9169  1.0907 

9495  1.0532 

9833  1.0170 

29 

32 

8551  1.1695 

8858  1.129C 

9174  1.0900 

9501  1.0526 

9838  1.0164 

28 

33 

8556  1.1688 

8863  1.1283 

9179  1.0894 

9506  1.0519 

9844  1.0158 

27 

34 

8561  1.1681 

8868  1.1276 

9185  1.0888 

9512  1.0513 

9850  1.0152 

26 

35 

8566  1.1674 

8873  1.1270 

9190  1.0881 

9517  1.0507 

9856  1.0147 

25 

36 

8571  1.1667 

8878  1.1263 

9195  1.0875 

9523  1.0501 

9861  1.0141 

24 

37 

8576  1.1660 

8884  1.1257 

9201  1.0869 

9528  1.0495 

9867  1.0135 

23 

38 

8581  1.1653 

8889  1.1250 

9206  1.0862 

9534  1.0489 

9873  1.0129 

22 

39 

8586  1.1647 

8894  1.1243 

9212  1.0856 

9540  1.0483 

9879  1.0123 

21 

40 

8591  1.1640 

8899  1:1237 

9217  1.0850 

9545  1.0477 

9884  1.0117 

20 

41 

8596  1.1633 

8904  1.1230 

9222  1.0843 

9551  1.0470 

9890  1.0111 

19 

42 

8601  1.1626 

8910  1.1224 

9228  1.0837 

9556  1.0464 

98%  1.0105 

18 

43 

8606  1.1619 

8915  1.1217 

9233  1.0831 

9562  1.0458 

9902  1.0099  17 

44 

8611  1.1612 

8920  1.1211 

9239  1.0824 

9567  1.0452 

9907  1.0094  !  16 

45 

8617  1.1606 

8925  1.1204 

9244  1.0818 

9573  1.0446 

.  9913  1.0088 

15 

46 

8622  1.1599 

8931  1.1197 

9249  1.0812 

9578  J.0440 

9919  1.0082 

14 

47 

8627  1.1592 

8936  1.1191 

9255  1.0805 

9584  1.0434 

9925  1.0076 

13 

48 

8632  1.1585 

8941  1.1184 

9260  1.0799 

9590  1.0428 

9930  1.0070 

12 

49 

8637  1.1578 

8946  1.1178 

9266  1.0793 

9595  1.0422 

9936  1.0064 

11 

5O 

8642  1.1571 

8952  1.1171 

9271  1.0786 

9601  1.0416 

9942  1.0058  1O 

51 

8647  1.1565 

8957  1.1165 

9276  1.0780 

9606  1.0410 

9948  1.0052 

9 

52 

8652  1.1558 

8962  1.1158 

9282  1.0774 

9612  1.0404 

9954  1.0047 

8 

53 

8657  1.1551 

8967  1.1152 

9287  1.0768 

9618  1.0398 

9959  1.0041 

7 

54 

8662  1.1544 

8972  1.1145 

9293  1.0761 

9623  1.0392 

9965  1.0035 

6 

55 

8667  1.1538 

8978  1.1139 

9298  1.0755 

9629  1.0385 

9971  1.0029 

5 

56 

8672  1.1531 

8983  1.1132 

9303  1.0749 

9634  1.0379 

9977  1.0023 

4 

57 

8678  1.1524 

8988  1.1126 

9309  1.0742 

9640  1.0373 

9983  1.0017 

3 

58 

8683  1.1517 

8994  1.1119 

9314  1.0736 

9646  1.0367 

9988  1.0012 

2 

59 

8688  1.1510 

8999  1.1113 

9320  1.0730 

9651  1.0361 

9994  1.0006 

1 

6O 

8693  1.1504 

9004  1.1106 

9325  1.0724 

9657  1.0355 

1.0000  1.0000 

0 

cot   tan 

cot   tan 

cot   tan 

cot    tan 

cot   tan 

F     49° 

48° 

47° 

46° 

45° 

/ 

INDEX. 


Agonic  line 201 

Alignment  in  mines 381-384 

obstacles  to  and  problems  on 17, 120, 304, 370 

street 360 

Angle  of  deflection  108 

of  depression  ;  of  elevation 56 

horizontal ;  vertical 66 

measurement  between  lines 112 

measurement  by  repetition 106 

measurement  by  series 107 

traversing  108 

problems 113 

Areas,  determination  of 35, 183, 187,  196 

Azimuth,  determination  by  solar  attachment 94 

table  of  errors  in 96 

Balancing  the  survey  161 

weights  used 166 

Bearings,  compass 102 

reverse 103 

proof 104 

changing 116 

when  the  variation  is  known 209 

to  obtain  true • 212 

Chain,  engineer's 4 

Gunter's ;  two-pole 3 

required  length  of  four-pole 301 

Chaining  4,  302,  334 

Circle,  area  of 33, 43 

laying  out 228-231, 388»  884 

vertical 388,389 

Clinometer,  the  hanging 390, 391 

Compass,  surveyor's 56 

adjustment 66 


82  INDEX. 

PAGE 

Compass  bearings 102 

solar  . . '. 276 

principles  and  adjustments  of  solar 280-285 

the  hanging 390 

the  diurnal  variation  of 200 

the  secular  variation  of 201,  208 

variation  determined  by  old  lines 210 

Contour  lines 393 

Co-ordinates,  application  of  rectangular 45 

Corners,  establishing 304 

lot 369 

quarter-section ;  witness 307 

re-establishment  of 316 

Course,  initial,  reduced 382 

special  appliances , 390 

Cross-sectioning,  angular 394 

Curvature  and  refraction 350,  351 

Curves,  how  laid  out 229,  230, 383,  384 

Declination,  allowance  for 281 

formula  expressing 203 

line  of  no 201 

of  the  needle 199 

by  Polaris  214 

problems  290,  291 

setting  off 285, 286 

Departure  and  latitude 154 

correcting 161 

Distances,  measurement  of  inaccessible 135-137 

Dividing  land 232 

Drainage 377 

Ellipse,  area  of 35 

laying  out 228-231 

Error  in  chaining 7, 8 

caused  by  needle 164 

the  survey 163 

Field  notes,  recording  the 22, 41, 142, 147, 318,  356, 385 

Formulas  expressing  magnetic  declination 203 

Grade  lines 357 

Grade  line  marking 366 

Grades,  determination  of 374 

Gradienter 100, 329 

Graduation  ..  .69 


INDEX.  83 

FAGK 

Heights,  measurement  of 20, 129-131 

Hexagon,  area  of  regular 31' 

Judicial  functions  of  surveyors 397 

Latitude  and  departure 154 

correcting 161 

difference  of 164 

finding  86 

by  circumpolar  star 92 

by  the  sun,  with  Saegmuller's  attachment 91 

setting  off 291 

Laying  out  land 221 

Level,  adjustment 340-343 

wye 337 

engraving  of  wye 338, 339 

use  of 343, 387, 388 

Locke's  hand 333 

rod 349 

Levelling-rods 345 

Levelling  defined 349 

fore  and  back  sights 352,  353 

general  method 354, 355 

Line,  agonic 201 

azimuth  of 65 

bearing  of 65 

base   308 

meridian  .    64 

meridian  distance  of 55 

obstacles  in 17, 122, 124,  126,  304 

inaccessibility  of  one  end 19, 126 

inaccessibility  of  both  ends .20, 127 

running  a  grade 357 

Lines,  offset  and  tie 44-61 

parallel   16 

perpendicular 13 


ranging 


in 


marking 303,  366 

preservation  of  points  in 361 

surveying  section 296 

surveying  township 298,  299,  301 

straightening  boundary 268 

variation  determined  by  old 210 

contour   393 


84  INDEX. 

PAGE 

Local  attraction 103 

Longitude,  difference  of 154 

Mapping,  see  Plotting. 

Meandering 307 

Measurement,  corrections 335 

suggestions  on  336 

obstacles  to,  problems  on 18, 124 

Meridian,  the  principal 309 

the  magnetic , 54 

auxiliary   310 

principal  base  and , 300 

establishing  with  transit 218 

establishing  with  solar  attachment 93 

obtaining  approximately   219 

convergency  of 324 

inclination  of 322,  323 

deflection  of  range  lines  from 324 

general  rule  for  obtaining  the  double 188 

Micrometer  or  stadia  wires 95,  96 

Needle,  the  magnetic 54, 68 

adjustment  of 59 

declination  of 199 

remagnetization  of 60 

Objects  to  be  noted  in  surveying  public  land 316 

Obstacles 17, 19, 120-127,  304,  370 

Octagon,  area  of  regular   31 

Omissions,  supplying 166 

Parallelogram,  area  of 31,  38, 183 

laying  out 225,  226 

Parallels,  standard 300,  309 

Perpendiculars  and  parallels,  problems  on 13, 15, 17, 120 

Pins,  marking 4, 302 

Plane-table  surveying 262 

Plane-table,  engraving  of 263 

adjustments    , 266 

surveying,  various  methods 267-269 

use  in  topography „ 393 

practical  suggestions  on 273 

exercises    274 

Plans 372 

Plot,  farm  153 

Plotting 26, 177, 180, 182,  385,  386 


INDEX.  85 

PAOB 

Poles,  or  rods,  for  sighting , 4f  331 

of  the  earth 55 

Polygon,  area  of 31-40, 186 

laying  out 227,  228 

dividing  up 253-258 

Profiles 363, 364,  373 

level  notes  for 356,  365 

Public  lands,  survey  of 275 

origin  of  system  of  survey 275,  295 

Record  of  levels  for  profile 365 

Recording  notes,  sfe  Field  Notes. 

Rectangle,  area  of 31,  37 

laying  out 224,  225 

Rectangular  surveying 297 

Refraction,  allowance  for 287 

tables 02, 288 

explanation  of  table  of 290 

problems  in 290, 291 

curvature  and 350 

Ring,  area  of 37, 43 

Rods  or  poles 4, 331 

levelling 345,  348 

Scale,  drawing  to 29 

to  adopt  30, 385 

unknown 30 

Scales 28 

Sections 299 

subdividing 314-316 

Segment,  area  of 34, 43 

Sextant,  area  of 34 

Shaft,  transference  of  points  down 389 

Solar  compass 276 

engraving  of 277 

hour  arc,  polar  axis 279 

principles 280 

adjustments .283-286 

to  use . 286 

running  lines  with 292 

time  for  using 296 

time  of  day  by  sun  with 294 

Solar  attachment  of  the  transit  (Gurley's)  79 

engraving  of 81 


86  INDEX. 

PAGE 

Solar  attachment,  adjustment  of 80 

running  lines  with 87 

Saegmuller's 88 

adjustment  of  Saegmuller's 88,  89 

time  and  azimuth  with  94 

Squares,  laying  out 223 

Stadia  wires,  or  micrometer 95,  96 

level  line  of  sight  perpendicular  to 97 

line  of  sight  inclined  to 99 

Streets,  direction  and  width  '. 358,  359 

alignment  of 360 

grade  and  profile 362,  363 

preservation  of  points  in  ...  361 

Table  of  areas  of  regular  polygons 33 

errors  in  azimuth 95 

elongation  and  azimuth  of  Polaris 216,  217 

inclination  and  convergency  of  meridians 325 

refraction  92 

secular  variation .  206 

traverse 156 


TABLES  AT  END  OF  BOOK. 

Tables  of  logarithms  of  numbers 1-19 

approximate  equations  of  time 20 

logarithms  of  trigonometric  functions . '. 21,  49 

for  determining  with  greater  accuracy 50,  51 

lengths  of  degree  of  latitude  and  longitude 52 

miscellaneous  formulae,  and  equivalents  of  metres,  chains,  and  feet,     53 

traverse 54-61 

natural  sines  and  cosines 62-70 

natural  tangents  and  cotangents 71-79 


Tallying 3 

Tape  measures 4,  332,  380 

Telescope,  section  of 65, 338,  339 

adjusting  level  on .... 78 

Testing  a  survey 160 

Time  by  solar  attachment 94 

Topography 362, 363,  391-393 

Township,  surveying  boundaries  of 298,  299,  301 

subdividing,.  ...311-313 


INDEX,  87 

PAQB 

Township  plan 290 

Transit,  showing  gradienter,  vertical  arc,  etc 329 

surveyor's  (full  circle) ' 64 

adjustments 71 

measurement  of  angles 106 

attachments   77 

adjustment  of  attachments  78 

solar  attachment  (see  Solar  Attachment) 79 

as  first  made 396 

to  take  apart , 79 

Trapezium,  area?  of 39, 184 

dividing 246-252 

Trapezoid,  area  of 31,  38, 184 

dividing 241-246 

Traverse  table 166 

Triangle,  area  of 31, 35, 183 

dividing 232-240 

laying  out J 221-223 

Tripod 71 

Tripods,  extra 381 

Vernier,  the 01 

the  transit 69 

determination  of  least  count  of 61 

to  read  an  instrument  having 62 

exercises  on  spacing 62 

Vertical  circle 77 

to  adjust  7V 


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